some reference requests for Borel algebras
$begingroup$
I believe the following are all true, and I could probably prove them myself if necessary. However this would be inelegant, and I would much prefer just to have references. I have been Googling quite a bit but no luck.
Let $E$ be a Lebesgue-measurable subset of $mathbb{R}$, and let $mathcal{B}=sigma(tau)$ be the Borel $sigma$-algebra on the usual metric topology $tau$ for $mathbb{R}$. Denote by $Ecapmathcal{B}$ the trace of $mathcal{B}$, i.e.
$$Ecapmathcal{B}={Ecap B:Binmathcal{B}}.$$
We also denote by $tau_E$ the subspace topology on $E$, i.e.
$$tau_E={Ecap U:Uintau}.$$
According to proofwiki, the following is true:
(1) $sigma(tau_E)=Ecapmathcal{B}$. In other words, the Borel $sigma$-algebra on $(E,tau_E)$ is just the family of all subsets of $E$ of the form $Ecap B$ for some Borel $Bsubseteqmathbb{R}$.
However, I need a "real" reference, i.e. something from a reputable publication.
Let $F$ be another Lebesgue-measurable subset of $mathbb{R}$, and suppose $f:Fto E$ is a bijective function which is strictly increasing. Then it is measurable in the following sense: If $cinmathbb{R}$ then ${xin F:f(x)<c}in Fcapmathcal{B}$.
(2) Is it true that $f^{-1}(A)in Fcapmathcal{B}$ for all $Ain Ecapmathcal{B}$? If (1) is true, this is equivalent to asking if $f^{-1}(A)insigma(tau_F)$ for all $Ainsigma(tau_E)$.
Finally, I have this one last reference request:
(3) Is it true that if $C$ is a Lebesgue-measurable subset of $mathbb{R}$ then there is a Borel set $Dinmathcal{B}$ such that $Csubseteq D$ and $Dsetminus C$ has Lebesgue measure zero?
I have heard that we can find $Ginmathcal{B}$ such that $CDelta G$ (the symmetric difference) has Lebesgue measure zero, and also that for any Lebesgue measure-zero set there exists a Borel set $Hinmathcal{B}$ of measure zero which contains it. If both of these facts are true then letting $D=Gcup H$ will do the trick for (3). But, again, I need references.
Thanks guys!
EDIT: I suppose if I can independently verify that these three things are all true, then I could just say something like "it is well-known that..." and see if the referee complains. But references would be ideal.
real-analysis reference-request lebesgue-measure borel-sets
asked Dec 12 '18 at 20:59
Ben WBen W
2,274615
$endgroup$
add a comment |
$begingroup$
I believe the following are all true, and I could probably prove them myself if necessary. However this would be inelegant, and I would much prefer just to have references. I have been Googling quite a bit but no luck.
Let $E$ be a Lebesgue-measurable subset of $mathbb{R}$, and let $mathcal{B}=sigma(tau)$ be the Borel $sigma$-algebra on the usual metric topology $tau$ for $mathbb{R}$. Denote by $Ecapmathcal{B}$ the trace of $mathcal{B}$, i.e.
$$Ecapmathcal{B}={Ecap B:Binmathcal{B}}.$$
We also denote by $tau_E$ the subspace topology on $E$, i.e.
$$tau_E={Ecap U:Uintau}.$$
According to proofwiki, the following is true:
(1) $sigma(tau_E)=Ecapmathcal{B}$. In other words, the Borel $sigma$-algebra on $(E,tau_E)$ is just the family of all subsets of $E$ of the form $Ecap B$ for some Borel $Bsubseteqmathbb{R}$.
However, I need a "real" reference, i.e. something from a reputable publication.
Let $F$ be another Lebesgue-measurable subset of $mathbb{R}$, and suppose $f:Fto E$ is a bijective function which is strictly increasing. Then it is measurable in the following sense: If $cinmathbb{R}$ then ${xin F:f(x)<c}in Fcapmathcal{B}$.
(2) Is it true that $f^{-1}(A)in Fcapmathcal{B}$ for all $Ain Ecapmathcal{B}$? If (1) is true, this is equivalent to asking if $f^{-1}(A)insigma(tau_F)$ for all $Ainsigma(tau_E)$.
Finally, I have this one last reference request:
(3) Is it true that if $C$ is a Lebesgue-measurable subset of $mathbb{R}$ then there is a Borel set $Dinmathcal{B}$ such that $Csubseteq D$ and $Dsetminus C$ has Lebesgue measure zero?
I have heard that we can find $Ginmathcal{B}$ such that $CDelta G$ (the symmetric difference) has Lebesgue measure zero, and also that for any Lebesgue measure-zero set there exists a Borel set $Hinmathcal{B}$ of measure zero which contains it. If both of these facts are true then letting $D=Gcup H$ will do the trick for (3). But, again, I need references.
Thanks guys!
EDIT: I suppose if I can independently verify that these three things are all true, then I could just say something like "it is well-known that..." and see if the referee complains. But references would be ideal.
real-analysis reference-request lebesgue-measure borel-sets
asked Dec 12 '18 at 20:59
Ben WBen W
2,274615
$endgroup$
add a comment |
$begingroup$
I believe the following are all true, and I could probably prove them myself if necessary. However this would be inelegant, and I would much prefer just to have references. I have been Googling quite a bit but no luck.
Let $E$ be a Lebesgue-measurable subset of $mathbb{R}$, and let $mathcal{B}=sigma(tau)$ be the Borel $sigma$-algebra on the usual metric topology $tau$ for $mathbb{R}$. Denote by $Ecapmathcal{B}$ the trace of $mathcal{B}$, i.e.
$$Ecapmathcal{B}={Ecap B:Binmathcal{B}}.$$
We also denote by $tau_E$ the subspace topology on $E$, i.e.
$$tau_E={Ecap U:Uintau}.$$
According to proofwiki, the following is true:
(1) $sigma(tau_E)=Ecapmathcal{B}$. In other words, the Borel $sigma$-algebra on $(E,tau_E)$ is just the family of all subsets of $E$ of the form $Ecap B$ for some Borel $Bsubseteqmathbb{R}$.
However, I need a "real" reference, i.e. something from a reputable publication.
Let $F$ be another Lebesgue-measurable subset of $mathbb{R}$, and suppose $f:Fto E$ is a bijective function which is strictly increasing. Then it is measurable in the following sense: If $cinmathbb{R}$ then ${xin F:f(x)<c}in Fcapmathcal{B}$.
(2) Is it true that $f^{-1}(A)in Fcapmathcal{B}$ for all $Ain Ecapmathcal{B}$? If (1) is true, this is equivalent to asking if $f^{-1}(A)insigma(tau_F)$ for all $Ainsigma(tau_E)$.
Finally, I have this one last reference request:
(3) Is it true that if $C$ is a Lebesgue-measurable subset of $mathbb{R}$ then there is a Borel set $Dinmathcal{B}$ such that $Csubseteq D$ and $Dsetminus C$ has Lebesgue measure zero?
I have heard that we can find $Ginmathcal{B}$ such that $CDelta G$ (the symmetric difference) has Lebesgue measure zero, and also that for any Lebesgue measure-zero set there exists a Borel set $Hinmathcal{B}$ of measure zero which contains it. If both of these facts are true then letting $D=Gcup H$ will do the trick for (3). But, again, I need references.
Thanks guys!
EDIT: I suppose if I can independently verify that these three things are all true, then I could just say something like "it is well-known that..." and see if the referee complains. But references would be ideal.
real-analysis reference-request lebesgue-measure borel-sets
asked Dec 12 '18 at 20:59
Ben WBen W
2,274615
$endgroup$
I believe the following are all true, and I could probably prove them myself if necessary. However this would be inelegant, and I would much prefer just to have references. I have been Googling quite a bit but no luck.
Let $E$ be a Lebesgue-measurable subset of $mathbb{R}$, and let $mathcal{B}=sigma(tau)$ be the Borel $sigma$-algebra on the usual metric topology $tau$ for $mathbb{R}$. Denote by $Ecapmathcal{B}$ the trace of $mathcal{B}$, i.e.
$$Ecapmathcal{B}={Ecap B:Binmathcal{B}}.$$
We also denote by $tau_E$ the subspace topology on $E$, i.e.
$$tau_E={Ecap U:Uintau}.$$
According to proofwiki, the following is true:
(1) $sigma(tau_E)=Ecapmathcal{B}$. In other words, the Borel $sigma$-algebra on $(E,tau_E)$ is just the family of all subsets of $E$ of the form $Ecap B$ for some Borel $Bsubseteqmathbb{R}$.
However, I need a "real" reference, i.e. something from a reputable publication.
Let $F$ be another Lebesgue-measurable subset of $mathbb{R}$, and suppose $f:Fto E$ is a bijective function which is strictly increasing. Then it is measurable in the following sense: If $cinmathbb{R}$ then ${xin F:f(x)<c}in Fcapmathcal{B}$.
(2) Is it true that $f^{-1}(A)in Fcapmathcal{B}$ for all $Ain Ecapmathcal{B}$? If (1) is true, this is equivalent to asking if $f^{-1}(A)insigma(tau_F)$ for all $Ainsigma(tau_E)$.
Finally, I have this one last reference request:
(3) Is it true that if $C$ is a Lebesgue-measurable subset of $mathbb{R}$ then there is a Borel set $Dinmathcal{B}$ such that $Csubseteq D$ and $Dsetminus C$ has Lebesgue measure zero?
I have heard that we can find $Ginmathcal{B}$ such that $CDelta G$ (the symmetric difference) has Lebesgue measure zero, and also that for any Lebesgue measure-zero set there exists a Borel set $Hinmathcal{B}$ of measure zero which contains it. If both of these facts are true then letting $D=Gcup H$ will do the trick for (3). But, again, I need references.
Thanks guys!
EDIT: I suppose if I can independently verify that these three things are all true, then I could just say something like "it is well-known that..." and see if the referee complains. But references would be ideal.
real-analysis reference-request lebesgue-measure borel-sets
real-analysis reference-request lebesgue-measure borel-sets
asked Dec 12 '18 at 20:59
Ben WBen W
2,274615
asked Dec 12 '18 at 20:59
Ben WBen W
2,274615
edited Dec 12 '18 at 21:01
Ben W
asked Dec 12 '18 at 20:59
Ben WBen W
2,274615
asked Dec 12 '18 at 20:59
Ben WBen W
2,274615
asked Dec 12 '18 at 20:59
Ben WBen W
2,274615
2,274615
add a comment |
add a comment |
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