If $int_0^x f dm$ is zero everywhere then $f$ is zero almost everywhere
I have been thinking on and off about a problem for some time now. It is inspired by an exam problem which I solved but I wanted to find an alternative solution. The object was to prove that some sequence of functions converges weakly to zero in $L^2$.
I managed to show (with some help) that the limit $f$ (of a subsequence) satisfies $int_0^x f dm=0$ for all $x>0 $. From this I want to conclude that $f=0$ a.e. I can do this with the fundamental theorem of calculus in its Lebesgue version but there ought to be a more elementary proof.
Can someone here help me out?
integration
add a comment |
I have been thinking on and off about a problem for some time now. It is inspired by an exam problem which I solved but I wanted to find an alternative solution. The object was to prove that some sequence of functions converges weakly to zero in $L^2$.
I managed to show (with some help) that the limit $f$ (of a subsequence) satisfies $int_0^x f dm=0$ for all $x>0 $. From this I want to conclude that $f=0$ a.e. I can do this with the fundamental theorem of calculus in its Lebesgue version but there ought to be a more elementary proof.
Can someone here help me out?
integration
add a comment |
I have been thinking on and off about a problem for some time now. It is inspired by an exam problem which I solved but I wanted to find an alternative solution. The object was to prove that some sequence of functions converges weakly to zero in $L^2$.
I managed to show (with some help) that the limit $f$ (of a subsequence) satisfies $int_0^x f dm=0$ for all $x>0 $. From this I want to conclude that $f=0$ a.e. I can do this with the fundamental theorem of calculus in its Lebesgue version but there ought to be a more elementary proof.
Can someone here help me out?
integration
I have been thinking on and off about a problem for some time now. It is inspired by an exam problem which I solved but I wanted to find an alternative solution. The object was to prove that some sequence of functions converges weakly to zero in $L^2$.
I managed to show (with some help) that the limit $f$ (of a subsequence) satisfies $int_0^x f dm=0$ for all $x>0 $. From this I want to conclude that $f=0$ a.e. I can do this with the fundamental theorem of calculus in its Lebesgue version but there ought to be a more elementary proof.
Can someone here help me out?
integration
integration
asked Jan 3 '11 at 16:22
JohanJohan
1,2011821
1,2011821
add a comment |
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5 Answers
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Indeed, as you expected, a simple proof of the result can be found; see Theorem 2.1 in this useful note on absolutely continuous functions.
EDIT: Since this is a quite important result, it is worth giving here the proof in detail. The proof below is essentially the one given in the link above, but somewhat shorter.
Theorem. If $f$ is integrable on $[a, b]$ and $int_a^x {f(t) dt} = 0$ $forall x in [a,b]$, then $f = 0$ a.e. on $[a, b]$.
Proof. An open subset $O$ of $[a,b]$ is a countable union of disjoint open intervals $(c_n, d_n)$; hence,
$$ int_O {f(t) dt} = sumlimits_{n = 1}^infty {int_{c_n }^{d_n } {f(t) dt} } = 0 $$.
If $K$ is a closed subset of $[a,b]$, then
$$ int_K {f(t) dt} = int_a^b f(t)dt - int_{(a, b) setminus K} f(t)dt = 0 - 0 = 0, $$
since $ (a, b) setminus K $ is open.
Next let $E_ + = { x in [a,b]:f(x) > 0}$ and $E_ - = { x in [a,b]:f(x) < 0}$. If $lambda(E_+) > 0$, then there exists some closed set $K subset E_+$ such that $lambda(K) > 0$. But $int_K {f(t){rm d}t} = 0$, hence $f=0$ a.e. on $K$. This contradiction shows that $lambda(E_+) = 0$. Similarly, $lambda(E_-) = 0$. The theorem is thus established.
2
$[a,b]setminus K$ is not necessarily open.
– nullUser
Oct 4 '12 at 1:47
add a comment |
It is sufficient to prove that $f$ is zero almost everywhere on any bounded interval.
(1) By additivity it is easy to see that $$int_a^bf(x)dx=int_0^bf(x)dx - int_0^af(x)dx$$ for all bounded intervals $(a,b)$ (and also for $[a,b)$, $(a,b]$ and $[a,b]$).
(2) Using (1) it is easy to see that $$int_Bf(x)dx=0$$ for any bounded Borel measurable set.
(3) Any Lebesgue measurable set $A$ is of the form $A=Bcup Z$ where $B$ is a Borel measurable set and $Z$ is a set of measure zero. Hence, by (2) we acheive
$$int_A f(x)dx= 0$$
for any bounded Lebesgue measurable set $A$.
(4) Now look at the sets $A_+(n)={x:f(x)>0}cap[-n,n]$ and $A_-(n)=[-n,n]setminus A_+(n)$. Assuming $f$ is measurable these sets are also measurable and by (3)
$$int_{A_pm(n)}f(x)dx=0$$
EDIT: and hence $f=0$ almost everywhere.
Please forgive me if I write $dx$ for the Lebesgue measure which I presume is what you refer to as $dm$.
Yes, that is what I mean by $dm$. Very nice argument.
– Johan
Jan 3 '11 at 23:00
How elementary is a proof of (3)?
– Aryabhata
Jan 4 '11 at 22:31
@Moron: Well, perhaps as elementary as the existence of the Lebesgue measure.
– AD.
Jan 5 '11 at 6:48
Never mind, I found a proof. It seems to be actually quite similar to my answer.
– Aryabhata
Jan 5 '11 at 7:16
1
@Moron: (2) is easier than that by (1) $int_I f dx=0$ for all bounded intervals $I$ and those intervals ($sigma-$)generates the bounded Borel sets.
– AD.
Jan 9 '11 at 7:44
|
show 2 more comments
I think you can use Dynkin's lemma (if you call this "more elementary").
Let D be all the measurable sets $Usubseteq I=[0,1]$ such that $intop_U f(t) = 0$ (the function $fmid_I$ is in $L_2$ so it is also in $L_1$, so I assume this from now). $Iin D$ and if $Asubseteq Bsubseteq I$ are in $D$ then $B-A in D$. If $A_i subseteq I$ is an increasing sequence in D then $bigcup A_i subseteq I$ is also in D (by the DCT). This shows that D is a Dynkin system.
Let P be all the open intervals in I (so $Psubseteq D$). P is not empty and an intersection of two open intervals are open, so P is closed under finite intersection, hence it is a pi system.
Dynkin's lemma says that if P is a pi system and D a dynkin system such that $Psubseteq D$ then $sigma(P)subseteq D$. The sigma algebra generated by P is the Borel algebra.
Now look on the set $A={xin I mid f(x)geq 0}$. This is a Lebesgue measurable set, so up to a zero measure set it is Borel measurable set $A'$. Since $intop_{A'} f(t) = 0$ and f is non negative there, then f is zero almost every where in A'. The same argument work for when f<0, so you get that f is zero almost everywhere in $I$. Now do this for all of $n+I,;nin mathbb{Z}$.
1
Very nice. The last step is unnecessary if $f$ is Borel measurable, and even if it is only Lebesgue measurable we could use the fact that a Lebesgue measurable function $f$ is a.e. equal to a Borel measurable function $tilde{f}$, and consider $tilde{f}$ instead.
– Nate Eldredge
Jan 3 '11 at 17:48
It was not obvious to me that the integral over a Borel set must be zero, I added a proof of that to my answer.
– Aryabhata
Jan 8 '11 at 23:10
add a comment |
I believe here is an elementary proof (if you are willing to call dominated convergence theorem as elementary).
First a lemma:
Lemma: let $displaystyle A$ be a bounded measurable set and let $displaystyle f in L(A)$. If $A_n subset A$ is a sequence of measurable sets such that
$$ A_1 supset A_2 supset A_3 supset dots$$
and $$lim_{n to infty} m(A_n) = 0$$
then
$$lim_{n to infty} int_{A_n} f text{dm} = 0$$
($displaystyle m(T)$ is the lebesgue measure of $displaystyle T$).
Proof:
It is well known (and has an elementary proof) that $displaystyle X = bigcap_{n=1}^{infty} A_n$ is measurable and $displaystyle m(X) = lim_{n to infty} m(A_n) = 0$.
Now define a sequence of (summable) functions
$displaystyle f_n(x) =
begin{cases} 2 f(x) & x in A_n \
f(x) & text{otherwise}
end{cases}$
Now $displaystyle |f_n(x)| le |2f(x)|$ and $f_n to f$ almost everywhere.
The set of points $displaystyle S$ where $f_n(x) to f(x)$ is not true, satisfies $displaystyle S subset X$ and hence is measurable and $displaystyle m(S) = 0$.
By the dominated convergence theorem we have that
$$lim_{n to infty} int_{A} f_n = int_{A} f$$
But we have that
$$int_{A} f_n = int_{A} f + int_{A_n} f$$
Thus
$$lim_{n to infty} int_{A_n} f = 0$$
$displaystyle circ$
Note that if $displaystyle f$ was bounded, then there is a much simpler proof of the above lemma, which does not make use of the dominated convergence theorem.
Now back to the original problem.
Let $displaystyle P_n = { x : f(x) ge frac{1}{n} }$.
If the set $displaystyle P = {x : f(x) gt 0} = bigcup P_n$ is of positive measure, then there is an $displaystyle n$ for which $displaystyle m(P_n) gt 0$. Now if $displaystyle P_n$ is unbounded, there is some $displaystyle M$ for which $displaystyle m(P_n cap [M, M+1]) gt 0$. Call that set $displaystyle A$.
Notice that $displaystyle int_{A} f ge frac{m(A)}{n} gt 0$.
Now give an integer $displaystyle k gt 0$, there is an open set $displaystyle G_k supset A$ such that $displaystyle m(G_k-A) lt frac{1}{k}$.
Note that we can choose the $displaystyle G_i$ such that $displaystyle G_1 supset G_2 supset G_3 supset dots$, by taking $displaystyle G'_k = bigcap_{i = 1}^{k} G_i$.
Now the sequence of sets $displaystyle A_k = G'_k -A$ satisfies the conditions of the above lemma,
we also have
$$int_{G'_k} f = int_{A} f + int_{A_k} f$$
Now since $displaystyle G'_{k}$ is a countable union of intervals, we have that $displaystyle int_{G'_k} f = 0$, since over every interval, the integral of $displaystyle f$ is $displaystyle 0$.
Thus
$$int_{A} f + int_{A_k} f = 0$$
Taking limits, and applying above lemma, we get
$$int_{A} f = 0$$
A contradiction. Similarly, we can show that negative set of $displaystyle f$ is of measure $displaystyle 0$ (or just consider $displaystyle -f$).
Hence $displaystyle f = 0 text{a.e}$
Note: Since this answer almost proves two claims made by other answers, I am including a sketch of proof of those here:
Claim 1) For any measurable set $displaystyle A$, there is a Borel Set $displaystyle B supset A$ such that $displaystyle m(B) = m(A)$.
For a proof of that, consider the $displaystyle G'_{k}$ above. $displaystyle B = bigcap_{k=1}^{infty} G'_{k}$ is a Borel set such that $displaystyle m(B) = m(A)$, as $displaystyle m(B) = lim_{k to infty} m(G'_{k}) = m(A)$.
Claim 2) For the $displaystyle f$ in the problem, for any Borel set $displaystyle B$, $displaystyle int_{B} f = 0$.
The proof above actually shows that for any measurable set $displaystyle E$, $displaystyle int_{E} f = 0$.
Nice!! But you forgot to mention (I think) how you make $P$ out of $P_n$.
– Jonas Teuwen
Jan 4 '11 at 19:21
@Jonas: Thanks! edited.
– Aryabhata
Jan 4 '11 at 19:58
Nice argument. But I do not understand your use of Lusin's theorem in the alternative solution. How do you conclude that $f$ must be continuous at some point? That the positive set has positive measure is not enough for general functions of course but I do not see how the fact that the function integrates to zero over intervals helps.
– Johan
Jan 5 '11 at 9:54
@Johan: I was mistaken, I missed the 'when restricted to" part of the theorem. In fact the characteristic function of rationals is a counterexample to what I had claimed. I have deleted that portion from the answer.
– Aryabhata
Jan 5 '11 at 19:18
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If $F(x)=int_0^x f(t) dt=0$ everywhere, then $F'(x)=0$ for all $x$. Since $f$ is locally integrable, $F'(x)=f(x)$ almost everywhere. Hence $f(x)=0$ almost everywhere.
This uses the fundamental theorem of calculus, which the OP said he wanted to avoid.
– Pete L. Clark
Jan 3 '11 at 16:33
You are right; I did not read closely. Since the OP is working with $L^2$, Lebesgue integration is unavoidable; I don't see the point of trying to avoid this basic theorem in the theoy.
– TCL
Jan 3 '11 at 16:39
3
The only point is that the fundamental theorem of calculus for Lebesgue integrals is treated quite late in the expositions I have seen, after one has proven a great deal of other properties of the integral. But the statement I am seeking to prove seems so simple and intuitive that there ought to be a simple proof.
– Johan
Jan 3 '11 at 22:55
add a comment |
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5 Answers
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Indeed, as you expected, a simple proof of the result can be found; see Theorem 2.1 in this useful note on absolutely continuous functions.
EDIT: Since this is a quite important result, it is worth giving here the proof in detail. The proof below is essentially the one given in the link above, but somewhat shorter.
Theorem. If $f$ is integrable on $[a, b]$ and $int_a^x {f(t) dt} = 0$ $forall x in [a,b]$, then $f = 0$ a.e. on $[a, b]$.
Proof. An open subset $O$ of $[a,b]$ is a countable union of disjoint open intervals $(c_n, d_n)$; hence,
$$ int_O {f(t) dt} = sumlimits_{n = 1}^infty {int_{c_n }^{d_n } {f(t) dt} } = 0 $$.
If $K$ is a closed subset of $[a,b]$, then
$$ int_K {f(t) dt} = int_a^b f(t)dt - int_{(a, b) setminus K} f(t)dt = 0 - 0 = 0, $$
since $ (a, b) setminus K $ is open.
Next let $E_ + = { x in [a,b]:f(x) > 0}$ and $E_ - = { x in [a,b]:f(x) < 0}$. If $lambda(E_+) > 0$, then there exists some closed set $K subset E_+$ such that $lambda(K) > 0$. But $int_K {f(t){rm d}t} = 0$, hence $f=0$ a.e. on $K$. This contradiction shows that $lambda(E_+) = 0$. Similarly, $lambda(E_-) = 0$. The theorem is thus established.
2
$[a,b]setminus K$ is not necessarily open.
– nullUser
Oct 4 '12 at 1:47
add a comment |
Indeed, as you expected, a simple proof of the result can be found; see Theorem 2.1 in this useful note on absolutely continuous functions.
EDIT: Since this is a quite important result, it is worth giving here the proof in detail. The proof below is essentially the one given in the link above, but somewhat shorter.
Theorem. If $f$ is integrable on $[a, b]$ and $int_a^x {f(t) dt} = 0$ $forall x in [a,b]$, then $f = 0$ a.e. on $[a, b]$.
Proof. An open subset $O$ of $[a,b]$ is a countable union of disjoint open intervals $(c_n, d_n)$; hence,
$$ int_O {f(t) dt} = sumlimits_{n = 1}^infty {int_{c_n }^{d_n } {f(t) dt} } = 0 $$.
If $K$ is a closed subset of $[a,b]$, then
$$ int_K {f(t) dt} = int_a^b f(t)dt - int_{(a, b) setminus K} f(t)dt = 0 - 0 = 0, $$
since $ (a, b) setminus K $ is open.
Next let $E_ + = { x in [a,b]:f(x) > 0}$ and $E_ - = { x in [a,b]:f(x) < 0}$. If $lambda(E_+) > 0$, then there exists some closed set $K subset E_+$ such that $lambda(K) > 0$. But $int_K {f(t){rm d}t} = 0$, hence $f=0$ a.e. on $K$. This contradiction shows that $lambda(E_+) = 0$. Similarly, $lambda(E_-) = 0$. The theorem is thus established.
2
$[a,b]setminus K$ is not necessarily open.
– nullUser
Oct 4 '12 at 1:47
add a comment |
Indeed, as you expected, a simple proof of the result can be found; see Theorem 2.1 in this useful note on absolutely continuous functions.
EDIT: Since this is a quite important result, it is worth giving here the proof in detail. The proof below is essentially the one given in the link above, but somewhat shorter.
Theorem. If $f$ is integrable on $[a, b]$ and $int_a^x {f(t) dt} = 0$ $forall x in [a,b]$, then $f = 0$ a.e. on $[a, b]$.
Proof. An open subset $O$ of $[a,b]$ is a countable union of disjoint open intervals $(c_n, d_n)$; hence,
$$ int_O {f(t) dt} = sumlimits_{n = 1}^infty {int_{c_n }^{d_n } {f(t) dt} } = 0 $$.
If $K$ is a closed subset of $[a,b]$, then
$$ int_K {f(t) dt} = int_a^b f(t)dt - int_{(a, b) setminus K} f(t)dt = 0 - 0 = 0, $$
since $ (a, b) setminus K $ is open.
Next let $E_ + = { x in [a,b]:f(x) > 0}$ and $E_ - = { x in [a,b]:f(x) < 0}$. If $lambda(E_+) > 0$, then there exists some closed set $K subset E_+$ such that $lambda(K) > 0$. But $int_K {f(t){rm d}t} = 0$, hence $f=0$ a.e. on $K$. This contradiction shows that $lambda(E_+) = 0$. Similarly, $lambda(E_-) = 0$. The theorem is thus established.
Indeed, as you expected, a simple proof of the result can be found; see Theorem 2.1 in this useful note on absolutely continuous functions.
EDIT: Since this is a quite important result, it is worth giving here the proof in detail. The proof below is essentially the one given in the link above, but somewhat shorter.
Theorem. If $f$ is integrable on $[a, b]$ and $int_a^x {f(t) dt} = 0$ $forall x in [a,b]$, then $f = 0$ a.e. on $[a, b]$.
Proof. An open subset $O$ of $[a,b]$ is a countable union of disjoint open intervals $(c_n, d_n)$; hence,
$$ int_O {f(t) dt} = sumlimits_{n = 1}^infty {int_{c_n }^{d_n } {f(t) dt} } = 0 $$.
If $K$ is a closed subset of $[a,b]$, then
$$ int_K {f(t) dt} = int_a^b f(t)dt - int_{(a, b) setminus K} f(t)dt = 0 - 0 = 0, $$
since $ (a, b) setminus K $ is open.
Next let $E_ + = { x in [a,b]:f(x) > 0}$ and $E_ - = { x in [a,b]:f(x) < 0}$. If $lambda(E_+) > 0$, then there exists some closed set $K subset E_+$ such that $lambda(K) > 0$. But $int_K {f(t){rm d}t} = 0$, hence $f=0$ a.e. on $K$. This contradiction shows that $lambda(E_+) = 0$. Similarly, $lambda(E_-) = 0$. The theorem is thus established.
edited Nov 29 '18 at 21:08
dasaphro
16512
16512
answered Jan 4 '11 at 6:09
Shai CovoShai Covo
21.2k23053
21.2k23053
2
$[a,b]setminus K$ is not necessarily open.
– nullUser
Oct 4 '12 at 1:47
add a comment |
2
$[a,b]setminus K$ is not necessarily open.
– nullUser
Oct 4 '12 at 1:47
2
2
$[a,b]setminus K$ is not necessarily open.
– nullUser
Oct 4 '12 at 1:47
$[a,b]setminus K$ is not necessarily open.
– nullUser
Oct 4 '12 at 1:47
add a comment |
It is sufficient to prove that $f$ is zero almost everywhere on any bounded interval.
(1) By additivity it is easy to see that $$int_a^bf(x)dx=int_0^bf(x)dx - int_0^af(x)dx$$ for all bounded intervals $(a,b)$ (and also for $[a,b)$, $(a,b]$ and $[a,b]$).
(2) Using (1) it is easy to see that $$int_Bf(x)dx=0$$ for any bounded Borel measurable set.
(3) Any Lebesgue measurable set $A$ is of the form $A=Bcup Z$ where $B$ is a Borel measurable set and $Z$ is a set of measure zero. Hence, by (2) we acheive
$$int_A f(x)dx= 0$$
for any bounded Lebesgue measurable set $A$.
(4) Now look at the sets $A_+(n)={x:f(x)>0}cap[-n,n]$ and $A_-(n)=[-n,n]setminus A_+(n)$. Assuming $f$ is measurable these sets are also measurable and by (3)
$$int_{A_pm(n)}f(x)dx=0$$
EDIT: and hence $f=0$ almost everywhere.
Please forgive me if I write $dx$ for the Lebesgue measure which I presume is what you refer to as $dm$.
Yes, that is what I mean by $dm$. Very nice argument.
– Johan
Jan 3 '11 at 23:00
How elementary is a proof of (3)?
– Aryabhata
Jan 4 '11 at 22:31
@Moron: Well, perhaps as elementary as the existence of the Lebesgue measure.
– AD.
Jan 5 '11 at 6:48
Never mind, I found a proof. It seems to be actually quite similar to my answer.
– Aryabhata
Jan 5 '11 at 7:16
1
@Moron: (2) is easier than that by (1) $int_I f dx=0$ for all bounded intervals $I$ and those intervals ($sigma-$)generates the bounded Borel sets.
– AD.
Jan 9 '11 at 7:44
|
show 2 more comments
It is sufficient to prove that $f$ is zero almost everywhere on any bounded interval.
(1) By additivity it is easy to see that $$int_a^bf(x)dx=int_0^bf(x)dx - int_0^af(x)dx$$ for all bounded intervals $(a,b)$ (and also for $[a,b)$, $(a,b]$ and $[a,b]$).
(2) Using (1) it is easy to see that $$int_Bf(x)dx=0$$ for any bounded Borel measurable set.
(3) Any Lebesgue measurable set $A$ is of the form $A=Bcup Z$ where $B$ is a Borel measurable set and $Z$ is a set of measure zero. Hence, by (2) we acheive
$$int_A f(x)dx= 0$$
for any bounded Lebesgue measurable set $A$.
(4) Now look at the sets $A_+(n)={x:f(x)>0}cap[-n,n]$ and $A_-(n)=[-n,n]setminus A_+(n)$. Assuming $f$ is measurable these sets are also measurable and by (3)
$$int_{A_pm(n)}f(x)dx=0$$
EDIT: and hence $f=0$ almost everywhere.
Please forgive me if I write $dx$ for the Lebesgue measure which I presume is what you refer to as $dm$.
Yes, that is what I mean by $dm$. Very nice argument.
– Johan
Jan 3 '11 at 23:00
How elementary is a proof of (3)?
– Aryabhata
Jan 4 '11 at 22:31
@Moron: Well, perhaps as elementary as the existence of the Lebesgue measure.
– AD.
Jan 5 '11 at 6:48
Never mind, I found a proof. It seems to be actually quite similar to my answer.
– Aryabhata
Jan 5 '11 at 7:16
1
@Moron: (2) is easier than that by (1) $int_I f dx=0$ for all bounded intervals $I$ and those intervals ($sigma-$)generates the bounded Borel sets.
– AD.
Jan 9 '11 at 7:44
|
show 2 more comments
It is sufficient to prove that $f$ is zero almost everywhere on any bounded interval.
(1) By additivity it is easy to see that $$int_a^bf(x)dx=int_0^bf(x)dx - int_0^af(x)dx$$ for all bounded intervals $(a,b)$ (and also for $[a,b)$, $(a,b]$ and $[a,b]$).
(2) Using (1) it is easy to see that $$int_Bf(x)dx=0$$ for any bounded Borel measurable set.
(3) Any Lebesgue measurable set $A$ is of the form $A=Bcup Z$ where $B$ is a Borel measurable set and $Z$ is a set of measure zero. Hence, by (2) we acheive
$$int_A f(x)dx= 0$$
for any bounded Lebesgue measurable set $A$.
(4) Now look at the sets $A_+(n)={x:f(x)>0}cap[-n,n]$ and $A_-(n)=[-n,n]setminus A_+(n)$. Assuming $f$ is measurable these sets are also measurable and by (3)
$$int_{A_pm(n)}f(x)dx=0$$
EDIT: and hence $f=0$ almost everywhere.
Please forgive me if I write $dx$ for the Lebesgue measure which I presume is what you refer to as $dm$.
It is sufficient to prove that $f$ is zero almost everywhere on any bounded interval.
(1) By additivity it is easy to see that $$int_a^bf(x)dx=int_0^bf(x)dx - int_0^af(x)dx$$ for all bounded intervals $(a,b)$ (and also for $[a,b)$, $(a,b]$ and $[a,b]$).
(2) Using (1) it is easy to see that $$int_Bf(x)dx=0$$ for any bounded Borel measurable set.
(3) Any Lebesgue measurable set $A$ is of the form $A=Bcup Z$ where $B$ is a Borel measurable set and $Z$ is a set of measure zero. Hence, by (2) we acheive
$$int_A f(x)dx= 0$$
for any bounded Lebesgue measurable set $A$.
(4) Now look at the sets $A_+(n)={x:f(x)>0}cap[-n,n]$ and $A_-(n)=[-n,n]setminus A_+(n)$. Assuming $f$ is measurable these sets are also measurable and by (3)
$$int_{A_pm(n)}f(x)dx=0$$
EDIT: and hence $f=0$ almost everywhere.
Please forgive me if I write $dx$ for the Lebesgue measure which I presume is what you refer to as $dm$.
edited Jan 4 '11 at 14:09
answered Jan 3 '11 at 21:34
AD.AD.
8,69383161
8,69383161
Yes, that is what I mean by $dm$. Very nice argument.
– Johan
Jan 3 '11 at 23:00
How elementary is a proof of (3)?
– Aryabhata
Jan 4 '11 at 22:31
@Moron: Well, perhaps as elementary as the existence of the Lebesgue measure.
– AD.
Jan 5 '11 at 6:48
Never mind, I found a proof. It seems to be actually quite similar to my answer.
– Aryabhata
Jan 5 '11 at 7:16
1
@Moron: (2) is easier than that by (1) $int_I f dx=0$ for all bounded intervals $I$ and those intervals ($sigma-$)generates the bounded Borel sets.
– AD.
Jan 9 '11 at 7:44
|
show 2 more comments
Yes, that is what I mean by $dm$. Very nice argument.
– Johan
Jan 3 '11 at 23:00
How elementary is a proof of (3)?
– Aryabhata
Jan 4 '11 at 22:31
@Moron: Well, perhaps as elementary as the existence of the Lebesgue measure.
– AD.
Jan 5 '11 at 6:48
Never mind, I found a proof. It seems to be actually quite similar to my answer.
– Aryabhata
Jan 5 '11 at 7:16
1
@Moron: (2) is easier than that by (1) $int_I f dx=0$ for all bounded intervals $I$ and those intervals ($sigma-$)generates the bounded Borel sets.
– AD.
Jan 9 '11 at 7:44
Yes, that is what I mean by $dm$. Very nice argument.
– Johan
Jan 3 '11 at 23:00
Yes, that is what I mean by $dm$. Very nice argument.
– Johan
Jan 3 '11 at 23:00
How elementary is a proof of (3)?
– Aryabhata
Jan 4 '11 at 22:31
How elementary is a proof of (3)?
– Aryabhata
Jan 4 '11 at 22:31
@Moron: Well, perhaps as elementary as the existence of the Lebesgue measure.
– AD.
Jan 5 '11 at 6:48
@Moron: Well, perhaps as elementary as the existence of the Lebesgue measure.
– AD.
Jan 5 '11 at 6:48
Never mind, I found a proof. It seems to be actually quite similar to my answer.
– Aryabhata
Jan 5 '11 at 7:16
Never mind, I found a proof. It seems to be actually quite similar to my answer.
– Aryabhata
Jan 5 '11 at 7:16
1
1
@Moron: (2) is easier than that by (1) $int_I f dx=0$ for all bounded intervals $I$ and those intervals ($sigma-$)generates the bounded Borel sets.
– AD.
Jan 9 '11 at 7:44
@Moron: (2) is easier than that by (1) $int_I f dx=0$ for all bounded intervals $I$ and those intervals ($sigma-$)generates the bounded Borel sets.
– AD.
Jan 9 '11 at 7:44
|
show 2 more comments
I think you can use Dynkin's lemma (if you call this "more elementary").
Let D be all the measurable sets $Usubseteq I=[0,1]$ such that $intop_U f(t) = 0$ (the function $fmid_I$ is in $L_2$ so it is also in $L_1$, so I assume this from now). $Iin D$ and if $Asubseteq Bsubseteq I$ are in $D$ then $B-A in D$. If $A_i subseteq I$ is an increasing sequence in D then $bigcup A_i subseteq I$ is also in D (by the DCT). This shows that D is a Dynkin system.
Let P be all the open intervals in I (so $Psubseteq D$). P is not empty and an intersection of two open intervals are open, so P is closed under finite intersection, hence it is a pi system.
Dynkin's lemma says that if P is a pi system and D a dynkin system such that $Psubseteq D$ then $sigma(P)subseteq D$. The sigma algebra generated by P is the Borel algebra.
Now look on the set $A={xin I mid f(x)geq 0}$. This is a Lebesgue measurable set, so up to a zero measure set it is Borel measurable set $A'$. Since $intop_{A'} f(t) = 0$ and f is non negative there, then f is zero almost every where in A'. The same argument work for when f<0, so you get that f is zero almost everywhere in $I$. Now do this for all of $n+I,;nin mathbb{Z}$.
1
Very nice. The last step is unnecessary if $f$ is Borel measurable, and even if it is only Lebesgue measurable we could use the fact that a Lebesgue measurable function $f$ is a.e. equal to a Borel measurable function $tilde{f}$, and consider $tilde{f}$ instead.
– Nate Eldredge
Jan 3 '11 at 17:48
It was not obvious to me that the integral over a Borel set must be zero, I added a proof of that to my answer.
– Aryabhata
Jan 8 '11 at 23:10
add a comment |
I think you can use Dynkin's lemma (if you call this "more elementary").
Let D be all the measurable sets $Usubseteq I=[0,1]$ such that $intop_U f(t) = 0$ (the function $fmid_I$ is in $L_2$ so it is also in $L_1$, so I assume this from now). $Iin D$ and if $Asubseteq Bsubseteq I$ are in $D$ then $B-A in D$. If $A_i subseteq I$ is an increasing sequence in D then $bigcup A_i subseteq I$ is also in D (by the DCT). This shows that D is a Dynkin system.
Let P be all the open intervals in I (so $Psubseteq D$). P is not empty and an intersection of two open intervals are open, so P is closed under finite intersection, hence it is a pi system.
Dynkin's lemma says that if P is a pi system and D a dynkin system such that $Psubseteq D$ then $sigma(P)subseteq D$. The sigma algebra generated by P is the Borel algebra.
Now look on the set $A={xin I mid f(x)geq 0}$. This is a Lebesgue measurable set, so up to a zero measure set it is Borel measurable set $A'$. Since $intop_{A'} f(t) = 0$ and f is non negative there, then f is zero almost every where in A'. The same argument work for when f<0, so you get that f is zero almost everywhere in $I$. Now do this for all of $n+I,;nin mathbb{Z}$.
1
Very nice. The last step is unnecessary if $f$ is Borel measurable, and even if it is only Lebesgue measurable we could use the fact that a Lebesgue measurable function $f$ is a.e. equal to a Borel measurable function $tilde{f}$, and consider $tilde{f}$ instead.
– Nate Eldredge
Jan 3 '11 at 17:48
It was not obvious to me that the integral over a Borel set must be zero, I added a proof of that to my answer.
– Aryabhata
Jan 8 '11 at 23:10
add a comment |
I think you can use Dynkin's lemma (if you call this "more elementary").
Let D be all the measurable sets $Usubseteq I=[0,1]$ such that $intop_U f(t) = 0$ (the function $fmid_I$ is in $L_2$ so it is also in $L_1$, so I assume this from now). $Iin D$ and if $Asubseteq Bsubseteq I$ are in $D$ then $B-A in D$. If $A_i subseteq I$ is an increasing sequence in D then $bigcup A_i subseteq I$ is also in D (by the DCT). This shows that D is a Dynkin system.
Let P be all the open intervals in I (so $Psubseteq D$). P is not empty and an intersection of two open intervals are open, so P is closed under finite intersection, hence it is a pi system.
Dynkin's lemma says that if P is a pi system and D a dynkin system such that $Psubseteq D$ then $sigma(P)subseteq D$. The sigma algebra generated by P is the Borel algebra.
Now look on the set $A={xin I mid f(x)geq 0}$. This is a Lebesgue measurable set, so up to a zero measure set it is Borel measurable set $A'$. Since $intop_{A'} f(t) = 0$ and f is non negative there, then f is zero almost every where in A'. The same argument work for when f<0, so you get that f is zero almost everywhere in $I$. Now do this for all of $n+I,;nin mathbb{Z}$.
I think you can use Dynkin's lemma (if you call this "more elementary").
Let D be all the measurable sets $Usubseteq I=[0,1]$ such that $intop_U f(t) = 0$ (the function $fmid_I$ is in $L_2$ so it is also in $L_1$, so I assume this from now). $Iin D$ and if $Asubseteq Bsubseteq I$ are in $D$ then $B-A in D$. If $A_i subseteq I$ is an increasing sequence in D then $bigcup A_i subseteq I$ is also in D (by the DCT). This shows that D is a Dynkin system.
Let P be all the open intervals in I (so $Psubseteq D$). P is not empty and an intersection of two open intervals are open, so P is closed under finite intersection, hence it is a pi system.
Dynkin's lemma says that if P is a pi system and D a dynkin system such that $Psubseteq D$ then $sigma(P)subseteq D$. The sigma algebra generated by P is the Borel algebra.
Now look on the set $A={xin I mid f(x)geq 0}$. This is a Lebesgue measurable set, so up to a zero measure set it is Borel measurable set $A'$. Since $intop_{A'} f(t) = 0$ and f is non negative there, then f is zero almost every where in A'. The same argument work for when f<0, so you get that f is zero almost everywhere in $I$. Now do this for all of $n+I,;nin mathbb{Z}$.
answered Jan 3 '11 at 17:30
OfirOfir
6,56012035
6,56012035
1
Very nice. The last step is unnecessary if $f$ is Borel measurable, and even if it is only Lebesgue measurable we could use the fact that a Lebesgue measurable function $f$ is a.e. equal to a Borel measurable function $tilde{f}$, and consider $tilde{f}$ instead.
– Nate Eldredge
Jan 3 '11 at 17:48
It was not obvious to me that the integral over a Borel set must be zero, I added a proof of that to my answer.
– Aryabhata
Jan 8 '11 at 23:10
add a comment |
1
Very nice. The last step is unnecessary if $f$ is Borel measurable, and even if it is only Lebesgue measurable we could use the fact that a Lebesgue measurable function $f$ is a.e. equal to a Borel measurable function $tilde{f}$, and consider $tilde{f}$ instead.
– Nate Eldredge
Jan 3 '11 at 17:48
It was not obvious to me that the integral over a Borel set must be zero, I added a proof of that to my answer.
– Aryabhata
Jan 8 '11 at 23:10
1
1
Very nice. The last step is unnecessary if $f$ is Borel measurable, and even if it is only Lebesgue measurable we could use the fact that a Lebesgue measurable function $f$ is a.e. equal to a Borel measurable function $tilde{f}$, and consider $tilde{f}$ instead.
– Nate Eldredge
Jan 3 '11 at 17:48
Very nice. The last step is unnecessary if $f$ is Borel measurable, and even if it is only Lebesgue measurable we could use the fact that a Lebesgue measurable function $f$ is a.e. equal to a Borel measurable function $tilde{f}$, and consider $tilde{f}$ instead.
– Nate Eldredge
Jan 3 '11 at 17:48
It was not obvious to me that the integral over a Borel set must be zero, I added a proof of that to my answer.
– Aryabhata
Jan 8 '11 at 23:10
It was not obvious to me that the integral over a Borel set must be zero, I added a proof of that to my answer.
– Aryabhata
Jan 8 '11 at 23:10
add a comment |
I believe here is an elementary proof (if you are willing to call dominated convergence theorem as elementary).
First a lemma:
Lemma: let $displaystyle A$ be a bounded measurable set and let $displaystyle f in L(A)$. If $A_n subset A$ is a sequence of measurable sets such that
$$ A_1 supset A_2 supset A_3 supset dots$$
and $$lim_{n to infty} m(A_n) = 0$$
then
$$lim_{n to infty} int_{A_n} f text{dm} = 0$$
($displaystyle m(T)$ is the lebesgue measure of $displaystyle T$).
Proof:
It is well known (and has an elementary proof) that $displaystyle X = bigcap_{n=1}^{infty} A_n$ is measurable and $displaystyle m(X) = lim_{n to infty} m(A_n) = 0$.
Now define a sequence of (summable) functions
$displaystyle f_n(x) =
begin{cases} 2 f(x) & x in A_n \
f(x) & text{otherwise}
end{cases}$
Now $displaystyle |f_n(x)| le |2f(x)|$ and $f_n to f$ almost everywhere.
The set of points $displaystyle S$ where $f_n(x) to f(x)$ is not true, satisfies $displaystyle S subset X$ and hence is measurable and $displaystyle m(S) = 0$.
By the dominated convergence theorem we have that
$$lim_{n to infty} int_{A} f_n = int_{A} f$$
But we have that
$$int_{A} f_n = int_{A} f + int_{A_n} f$$
Thus
$$lim_{n to infty} int_{A_n} f = 0$$
$displaystyle circ$
Note that if $displaystyle f$ was bounded, then there is a much simpler proof of the above lemma, which does not make use of the dominated convergence theorem.
Now back to the original problem.
Let $displaystyle P_n = { x : f(x) ge frac{1}{n} }$.
If the set $displaystyle P = {x : f(x) gt 0} = bigcup P_n$ is of positive measure, then there is an $displaystyle n$ for which $displaystyle m(P_n) gt 0$. Now if $displaystyle P_n$ is unbounded, there is some $displaystyle M$ for which $displaystyle m(P_n cap [M, M+1]) gt 0$. Call that set $displaystyle A$.
Notice that $displaystyle int_{A} f ge frac{m(A)}{n} gt 0$.
Now give an integer $displaystyle k gt 0$, there is an open set $displaystyle G_k supset A$ such that $displaystyle m(G_k-A) lt frac{1}{k}$.
Note that we can choose the $displaystyle G_i$ such that $displaystyle G_1 supset G_2 supset G_3 supset dots$, by taking $displaystyle G'_k = bigcap_{i = 1}^{k} G_i$.
Now the sequence of sets $displaystyle A_k = G'_k -A$ satisfies the conditions of the above lemma,
we also have
$$int_{G'_k} f = int_{A} f + int_{A_k} f$$
Now since $displaystyle G'_{k}$ is a countable union of intervals, we have that $displaystyle int_{G'_k} f = 0$, since over every interval, the integral of $displaystyle f$ is $displaystyle 0$.
Thus
$$int_{A} f + int_{A_k} f = 0$$
Taking limits, and applying above lemma, we get
$$int_{A} f = 0$$
A contradiction. Similarly, we can show that negative set of $displaystyle f$ is of measure $displaystyle 0$ (or just consider $displaystyle -f$).
Hence $displaystyle f = 0 text{a.e}$
Note: Since this answer almost proves two claims made by other answers, I am including a sketch of proof of those here:
Claim 1) For any measurable set $displaystyle A$, there is a Borel Set $displaystyle B supset A$ such that $displaystyle m(B) = m(A)$.
For a proof of that, consider the $displaystyle G'_{k}$ above. $displaystyle B = bigcap_{k=1}^{infty} G'_{k}$ is a Borel set such that $displaystyle m(B) = m(A)$, as $displaystyle m(B) = lim_{k to infty} m(G'_{k}) = m(A)$.
Claim 2) For the $displaystyle f$ in the problem, for any Borel set $displaystyle B$, $displaystyle int_{B} f = 0$.
The proof above actually shows that for any measurable set $displaystyle E$, $displaystyle int_{E} f = 0$.
Nice!! But you forgot to mention (I think) how you make $P$ out of $P_n$.
– Jonas Teuwen
Jan 4 '11 at 19:21
@Jonas: Thanks! edited.
– Aryabhata
Jan 4 '11 at 19:58
Nice argument. But I do not understand your use of Lusin's theorem in the alternative solution. How do you conclude that $f$ must be continuous at some point? That the positive set has positive measure is not enough for general functions of course but I do not see how the fact that the function integrates to zero over intervals helps.
– Johan
Jan 5 '11 at 9:54
@Johan: I was mistaken, I missed the 'when restricted to" part of the theorem. In fact the characteristic function of rationals is a counterexample to what I had claimed. I have deleted that portion from the answer.
– Aryabhata
Jan 5 '11 at 19:18
add a comment |
I believe here is an elementary proof (if you are willing to call dominated convergence theorem as elementary).
First a lemma:
Lemma: let $displaystyle A$ be a bounded measurable set and let $displaystyle f in L(A)$. If $A_n subset A$ is a sequence of measurable sets such that
$$ A_1 supset A_2 supset A_3 supset dots$$
and $$lim_{n to infty} m(A_n) = 0$$
then
$$lim_{n to infty} int_{A_n} f text{dm} = 0$$
($displaystyle m(T)$ is the lebesgue measure of $displaystyle T$).
Proof:
It is well known (and has an elementary proof) that $displaystyle X = bigcap_{n=1}^{infty} A_n$ is measurable and $displaystyle m(X) = lim_{n to infty} m(A_n) = 0$.
Now define a sequence of (summable) functions
$displaystyle f_n(x) =
begin{cases} 2 f(x) & x in A_n \
f(x) & text{otherwise}
end{cases}$
Now $displaystyle |f_n(x)| le |2f(x)|$ and $f_n to f$ almost everywhere.
The set of points $displaystyle S$ where $f_n(x) to f(x)$ is not true, satisfies $displaystyle S subset X$ and hence is measurable and $displaystyle m(S) = 0$.
By the dominated convergence theorem we have that
$$lim_{n to infty} int_{A} f_n = int_{A} f$$
But we have that
$$int_{A} f_n = int_{A} f + int_{A_n} f$$
Thus
$$lim_{n to infty} int_{A_n} f = 0$$
$displaystyle circ$
Note that if $displaystyle f$ was bounded, then there is a much simpler proof of the above lemma, which does not make use of the dominated convergence theorem.
Now back to the original problem.
Let $displaystyle P_n = { x : f(x) ge frac{1}{n} }$.
If the set $displaystyle P = {x : f(x) gt 0} = bigcup P_n$ is of positive measure, then there is an $displaystyle n$ for which $displaystyle m(P_n) gt 0$. Now if $displaystyle P_n$ is unbounded, there is some $displaystyle M$ for which $displaystyle m(P_n cap [M, M+1]) gt 0$. Call that set $displaystyle A$.
Notice that $displaystyle int_{A} f ge frac{m(A)}{n} gt 0$.
Now give an integer $displaystyle k gt 0$, there is an open set $displaystyle G_k supset A$ such that $displaystyle m(G_k-A) lt frac{1}{k}$.
Note that we can choose the $displaystyle G_i$ such that $displaystyle G_1 supset G_2 supset G_3 supset dots$, by taking $displaystyle G'_k = bigcap_{i = 1}^{k} G_i$.
Now the sequence of sets $displaystyle A_k = G'_k -A$ satisfies the conditions of the above lemma,
we also have
$$int_{G'_k} f = int_{A} f + int_{A_k} f$$
Now since $displaystyle G'_{k}$ is a countable union of intervals, we have that $displaystyle int_{G'_k} f = 0$, since over every interval, the integral of $displaystyle f$ is $displaystyle 0$.
Thus
$$int_{A} f + int_{A_k} f = 0$$
Taking limits, and applying above lemma, we get
$$int_{A} f = 0$$
A contradiction. Similarly, we can show that negative set of $displaystyle f$ is of measure $displaystyle 0$ (or just consider $displaystyle -f$).
Hence $displaystyle f = 0 text{a.e}$
Note: Since this answer almost proves two claims made by other answers, I am including a sketch of proof of those here:
Claim 1) For any measurable set $displaystyle A$, there is a Borel Set $displaystyle B supset A$ such that $displaystyle m(B) = m(A)$.
For a proof of that, consider the $displaystyle G'_{k}$ above. $displaystyle B = bigcap_{k=1}^{infty} G'_{k}$ is a Borel set such that $displaystyle m(B) = m(A)$, as $displaystyle m(B) = lim_{k to infty} m(G'_{k}) = m(A)$.
Claim 2) For the $displaystyle f$ in the problem, for any Borel set $displaystyle B$, $displaystyle int_{B} f = 0$.
The proof above actually shows that for any measurable set $displaystyle E$, $displaystyle int_{E} f = 0$.
Nice!! But you forgot to mention (I think) how you make $P$ out of $P_n$.
– Jonas Teuwen
Jan 4 '11 at 19:21
@Jonas: Thanks! edited.
– Aryabhata
Jan 4 '11 at 19:58
Nice argument. But I do not understand your use of Lusin's theorem in the alternative solution. How do you conclude that $f$ must be continuous at some point? That the positive set has positive measure is not enough for general functions of course but I do not see how the fact that the function integrates to zero over intervals helps.
– Johan
Jan 5 '11 at 9:54
@Johan: I was mistaken, I missed the 'when restricted to" part of the theorem. In fact the characteristic function of rationals is a counterexample to what I had claimed. I have deleted that portion from the answer.
– Aryabhata
Jan 5 '11 at 19:18
add a comment |
I believe here is an elementary proof (if you are willing to call dominated convergence theorem as elementary).
First a lemma:
Lemma: let $displaystyle A$ be a bounded measurable set and let $displaystyle f in L(A)$. If $A_n subset A$ is a sequence of measurable sets such that
$$ A_1 supset A_2 supset A_3 supset dots$$
and $$lim_{n to infty} m(A_n) = 0$$
then
$$lim_{n to infty} int_{A_n} f text{dm} = 0$$
($displaystyle m(T)$ is the lebesgue measure of $displaystyle T$).
Proof:
It is well known (and has an elementary proof) that $displaystyle X = bigcap_{n=1}^{infty} A_n$ is measurable and $displaystyle m(X) = lim_{n to infty} m(A_n) = 0$.
Now define a sequence of (summable) functions
$displaystyle f_n(x) =
begin{cases} 2 f(x) & x in A_n \
f(x) & text{otherwise}
end{cases}$
Now $displaystyle |f_n(x)| le |2f(x)|$ and $f_n to f$ almost everywhere.
The set of points $displaystyle S$ where $f_n(x) to f(x)$ is not true, satisfies $displaystyle S subset X$ and hence is measurable and $displaystyle m(S) = 0$.
By the dominated convergence theorem we have that
$$lim_{n to infty} int_{A} f_n = int_{A} f$$
But we have that
$$int_{A} f_n = int_{A} f + int_{A_n} f$$
Thus
$$lim_{n to infty} int_{A_n} f = 0$$
$displaystyle circ$
Note that if $displaystyle f$ was bounded, then there is a much simpler proof of the above lemma, which does not make use of the dominated convergence theorem.
Now back to the original problem.
Let $displaystyle P_n = { x : f(x) ge frac{1}{n} }$.
If the set $displaystyle P = {x : f(x) gt 0} = bigcup P_n$ is of positive measure, then there is an $displaystyle n$ for which $displaystyle m(P_n) gt 0$. Now if $displaystyle P_n$ is unbounded, there is some $displaystyle M$ for which $displaystyle m(P_n cap [M, M+1]) gt 0$. Call that set $displaystyle A$.
Notice that $displaystyle int_{A} f ge frac{m(A)}{n} gt 0$.
Now give an integer $displaystyle k gt 0$, there is an open set $displaystyle G_k supset A$ such that $displaystyle m(G_k-A) lt frac{1}{k}$.
Note that we can choose the $displaystyle G_i$ such that $displaystyle G_1 supset G_2 supset G_3 supset dots$, by taking $displaystyle G'_k = bigcap_{i = 1}^{k} G_i$.
Now the sequence of sets $displaystyle A_k = G'_k -A$ satisfies the conditions of the above lemma,
we also have
$$int_{G'_k} f = int_{A} f + int_{A_k} f$$
Now since $displaystyle G'_{k}$ is a countable union of intervals, we have that $displaystyle int_{G'_k} f = 0$, since over every interval, the integral of $displaystyle f$ is $displaystyle 0$.
Thus
$$int_{A} f + int_{A_k} f = 0$$
Taking limits, and applying above lemma, we get
$$int_{A} f = 0$$
A contradiction. Similarly, we can show that negative set of $displaystyle f$ is of measure $displaystyle 0$ (or just consider $displaystyle -f$).
Hence $displaystyle f = 0 text{a.e}$
Note: Since this answer almost proves two claims made by other answers, I am including a sketch of proof of those here:
Claim 1) For any measurable set $displaystyle A$, there is a Borel Set $displaystyle B supset A$ such that $displaystyle m(B) = m(A)$.
For a proof of that, consider the $displaystyle G'_{k}$ above. $displaystyle B = bigcap_{k=1}^{infty} G'_{k}$ is a Borel set such that $displaystyle m(B) = m(A)$, as $displaystyle m(B) = lim_{k to infty} m(G'_{k}) = m(A)$.
Claim 2) For the $displaystyle f$ in the problem, for any Borel set $displaystyle B$, $displaystyle int_{B} f = 0$.
The proof above actually shows that for any measurable set $displaystyle E$, $displaystyle int_{E} f = 0$.
I believe here is an elementary proof (if you are willing to call dominated convergence theorem as elementary).
First a lemma:
Lemma: let $displaystyle A$ be a bounded measurable set and let $displaystyle f in L(A)$. If $A_n subset A$ is a sequence of measurable sets such that
$$ A_1 supset A_2 supset A_3 supset dots$$
and $$lim_{n to infty} m(A_n) = 0$$
then
$$lim_{n to infty} int_{A_n} f text{dm} = 0$$
($displaystyle m(T)$ is the lebesgue measure of $displaystyle T$).
Proof:
It is well known (and has an elementary proof) that $displaystyle X = bigcap_{n=1}^{infty} A_n$ is measurable and $displaystyle m(X) = lim_{n to infty} m(A_n) = 0$.
Now define a sequence of (summable) functions
$displaystyle f_n(x) =
begin{cases} 2 f(x) & x in A_n \
f(x) & text{otherwise}
end{cases}$
Now $displaystyle |f_n(x)| le |2f(x)|$ and $f_n to f$ almost everywhere.
The set of points $displaystyle S$ where $f_n(x) to f(x)$ is not true, satisfies $displaystyle S subset X$ and hence is measurable and $displaystyle m(S) = 0$.
By the dominated convergence theorem we have that
$$lim_{n to infty} int_{A} f_n = int_{A} f$$
But we have that
$$int_{A} f_n = int_{A} f + int_{A_n} f$$
Thus
$$lim_{n to infty} int_{A_n} f = 0$$
$displaystyle circ$
Note that if $displaystyle f$ was bounded, then there is a much simpler proof of the above lemma, which does not make use of the dominated convergence theorem.
Now back to the original problem.
Let $displaystyle P_n = { x : f(x) ge frac{1}{n} }$.
If the set $displaystyle P = {x : f(x) gt 0} = bigcup P_n$ is of positive measure, then there is an $displaystyle n$ for which $displaystyle m(P_n) gt 0$. Now if $displaystyle P_n$ is unbounded, there is some $displaystyle M$ for which $displaystyle m(P_n cap [M, M+1]) gt 0$. Call that set $displaystyle A$.
Notice that $displaystyle int_{A} f ge frac{m(A)}{n} gt 0$.
Now give an integer $displaystyle k gt 0$, there is an open set $displaystyle G_k supset A$ such that $displaystyle m(G_k-A) lt frac{1}{k}$.
Note that we can choose the $displaystyle G_i$ such that $displaystyle G_1 supset G_2 supset G_3 supset dots$, by taking $displaystyle G'_k = bigcap_{i = 1}^{k} G_i$.
Now the sequence of sets $displaystyle A_k = G'_k -A$ satisfies the conditions of the above lemma,
we also have
$$int_{G'_k} f = int_{A} f + int_{A_k} f$$
Now since $displaystyle G'_{k}$ is a countable union of intervals, we have that $displaystyle int_{G'_k} f = 0$, since over every interval, the integral of $displaystyle f$ is $displaystyle 0$.
Thus
$$int_{A} f + int_{A_k} f = 0$$
Taking limits, and applying above lemma, we get
$$int_{A} f = 0$$
A contradiction. Similarly, we can show that negative set of $displaystyle f$ is of measure $displaystyle 0$ (or just consider $displaystyle -f$).
Hence $displaystyle f = 0 text{a.e}$
Note: Since this answer almost proves two claims made by other answers, I am including a sketch of proof of those here:
Claim 1) For any measurable set $displaystyle A$, there is a Borel Set $displaystyle B supset A$ such that $displaystyle m(B) = m(A)$.
For a proof of that, consider the $displaystyle G'_{k}$ above. $displaystyle B = bigcap_{k=1}^{infty} G'_{k}$ is a Borel set such that $displaystyle m(B) = m(A)$, as $displaystyle m(B) = lim_{k to infty} m(G'_{k}) = m(A)$.
Claim 2) For the $displaystyle f$ in the problem, for any Borel set $displaystyle B$, $displaystyle int_{B} f = 0$.
The proof above actually shows that for any measurable set $displaystyle E$, $displaystyle int_{E} f = 0$.
edited Jan 9 '11 at 12:38
answered Jan 4 '11 at 18:55
AryabhataAryabhata
70.1k6156246
70.1k6156246
Nice!! But you forgot to mention (I think) how you make $P$ out of $P_n$.
– Jonas Teuwen
Jan 4 '11 at 19:21
@Jonas: Thanks! edited.
– Aryabhata
Jan 4 '11 at 19:58
Nice argument. But I do not understand your use of Lusin's theorem in the alternative solution. How do you conclude that $f$ must be continuous at some point? That the positive set has positive measure is not enough for general functions of course but I do not see how the fact that the function integrates to zero over intervals helps.
– Johan
Jan 5 '11 at 9:54
@Johan: I was mistaken, I missed the 'when restricted to" part of the theorem. In fact the characteristic function of rationals is a counterexample to what I had claimed. I have deleted that portion from the answer.
– Aryabhata
Jan 5 '11 at 19:18
add a comment |
Nice!! But you forgot to mention (I think) how you make $P$ out of $P_n$.
– Jonas Teuwen
Jan 4 '11 at 19:21
@Jonas: Thanks! edited.
– Aryabhata
Jan 4 '11 at 19:58
Nice argument. But I do not understand your use of Lusin's theorem in the alternative solution. How do you conclude that $f$ must be continuous at some point? That the positive set has positive measure is not enough for general functions of course but I do not see how the fact that the function integrates to zero over intervals helps.
– Johan
Jan 5 '11 at 9:54
@Johan: I was mistaken, I missed the 'when restricted to" part of the theorem. In fact the characteristic function of rationals is a counterexample to what I had claimed. I have deleted that portion from the answer.
– Aryabhata
Jan 5 '11 at 19:18
Nice!! But you forgot to mention (I think) how you make $P$ out of $P_n$.
– Jonas Teuwen
Jan 4 '11 at 19:21
Nice!! But you forgot to mention (I think) how you make $P$ out of $P_n$.
– Jonas Teuwen
Jan 4 '11 at 19:21
@Jonas: Thanks! edited.
– Aryabhata
Jan 4 '11 at 19:58
@Jonas: Thanks! edited.
– Aryabhata
Jan 4 '11 at 19:58
Nice argument. But I do not understand your use of Lusin's theorem in the alternative solution. How do you conclude that $f$ must be continuous at some point? That the positive set has positive measure is not enough for general functions of course but I do not see how the fact that the function integrates to zero over intervals helps.
– Johan
Jan 5 '11 at 9:54
Nice argument. But I do not understand your use of Lusin's theorem in the alternative solution. How do you conclude that $f$ must be continuous at some point? That the positive set has positive measure is not enough for general functions of course but I do not see how the fact that the function integrates to zero over intervals helps.
– Johan
Jan 5 '11 at 9:54
@Johan: I was mistaken, I missed the 'when restricted to" part of the theorem. In fact the characteristic function of rationals is a counterexample to what I had claimed. I have deleted that portion from the answer.
– Aryabhata
Jan 5 '11 at 19:18
@Johan: I was mistaken, I missed the 'when restricted to" part of the theorem. In fact the characteristic function of rationals is a counterexample to what I had claimed. I have deleted that portion from the answer.
– Aryabhata
Jan 5 '11 at 19:18
add a comment |
If $F(x)=int_0^x f(t) dt=0$ everywhere, then $F'(x)=0$ for all $x$. Since $f$ is locally integrable, $F'(x)=f(x)$ almost everywhere. Hence $f(x)=0$ almost everywhere.
This uses the fundamental theorem of calculus, which the OP said he wanted to avoid.
– Pete L. Clark
Jan 3 '11 at 16:33
You are right; I did not read closely. Since the OP is working with $L^2$, Lebesgue integration is unavoidable; I don't see the point of trying to avoid this basic theorem in the theoy.
– TCL
Jan 3 '11 at 16:39
3
The only point is that the fundamental theorem of calculus for Lebesgue integrals is treated quite late in the expositions I have seen, after one has proven a great deal of other properties of the integral. But the statement I am seeking to prove seems so simple and intuitive that there ought to be a simple proof.
– Johan
Jan 3 '11 at 22:55
add a comment |
If $F(x)=int_0^x f(t) dt=0$ everywhere, then $F'(x)=0$ for all $x$. Since $f$ is locally integrable, $F'(x)=f(x)$ almost everywhere. Hence $f(x)=0$ almost everywhere.
This uses the fundamental theorem of calculus, which the OP said he wanted to avoid.
– Pete L. Clark
Jan 3 '11 at 16:33
You are right; I did not read closely. Since the OP is working with $L^2$, Lebesgue integration is unavoidable; I don't see the point of trying to avoid this basic theorem in the theoy.
– TCL
Jan 3 '11 at 16:39
3
The only point is that the fundamental theorem of calculus for Lebesgue integrals is treated quite late in the expositions I have seen, after one has proven a great deal of other properties of the integral. But the statement I am seeking to prove seems so simple and intuitive that there ought to be a simple proof.
– Johan
Jan 3 '11 at 22:55
add a comment |
If $F(x)=int_0^x f(t) dt=0$ everywhere, then $F'(x)=0$ for all $x$. Since $f$ is locally integrable, $F'(x)=f(x)$ almost everywhere. Hence $f(x)=0$ almost everywhere.
If $F(x)=int_0^x f(t) dt=0$ everywhere, then $F'(x)=0$ for all $x$. Since $f$ is locally integrable, $F'(x)=f(x)$ almost everywhere. Hence $f(x)=0$ almost everywhere.
answered Jan 3 '11 at 16:28
TCLTCL
9,13952058
9,13952058
This uses the fundamental theorem of calculus, which the OP said he wanted to avoid.
– Pete L. Clark
Jan 3 '11 at 16:33
You are right; I did not read closely. Since the OP is working with $L^2$, Lebesgue integration is unavoidable; I don't see the point of trying to avoid this basic theorem in the theoy.
– TCL
Jan 3 '11 at 16:39
3
The only point is that the fundamental theorem of calculus for Lebesgue integrals is treated quite late in the expositions I have seen, after one has proven a great deal of other properties of the integral. But the statement I am seeking to prove seems so simple and intuitive that there ought to be a simple proof.
– Johan
Jan 3 '11 at 22:55
add a comment |
This uses the fundamental theorem of calculus, which the OP said he wanted to avoid.
– Pete L. Clark
Jan 3 '11 at 16:33
You are right; I did not read closely. Since the OP is working with $L^2$, Lebesgue integration is unavoidable; I don't see the point of trying to avoid this basic theorem in the theoy.
– TCL
Jan 3 '11 at 16:39
3
The only point is that the fundamental theorem of calculus for Lebesgue integrals is treated quite late in the expositions I have seen, after one has proven a great deal of other properties of the integral. But the statement I am seeking to prove seems so simple and intuitive that there ought to be a simple proof.
– Johan
Jan 3 '11 at 22:55
This uses the fundamental theorem of calculus, which the OP said he wanted to avoid.
– Pete L. Clark
Jan 3 '11 at 16:33
This uses the fundamental theorem of calculus, which the OP said he wanted to avoid.
– Pete L. Clark
Jan 3 '11 at 16:33
You are right; I did not read closely. Since the OP is working with $L^2$, Lebesgue integration is unavoidable; I don't see the point of trying to avoid this basic theorem in the theoy.
– TCL
Jan 3 '11 at 16:39
You are right; I did not read closely. Since the OP is working with $L^2$, Lebesgue integration is unavoidable; I don't see the point of trying to avoid this basic theorem in the theoy.
– TCL
Jan 3 '11 at 16:39
3
3
The only point is that the fundamental theorem of calculus for Lebesgue integrals is treated quite late in the expositions I have seen, after one has proven a great deal of other properties of the integral. But the statement I am seeking to prove seems so simple and intuitive that there ought to be a simple proof.
– Johan
Jan 3 '11 at 22:55
The only point is that the fundamental theorem of calculus for Lebesgue integrals is treated quite late in the expositions I have seen, after one has proven a great deal of other properties of the integral. But the statement I am seeking to prove seems so simple and intuitive that there ought to be a simple proof.
– Johan
Jan 3 '11 at 22:55
add a comment |
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