Image of measure is relatively compact
$begingroup$
Let $B$ be a Banach space and $mu$ a $B$-valued measure on $X$. If $f: X to B$ is a $mu$-integrable function, then define
$$mu_f(E) = int_E f dmu$$
Is it true that the image of $mu_f$ is relatively compact in $B$? I guess the real question is: if $mu$ is totally finite measure, is it true that the image of $mu$ is relatively compact?
@below
You are right. Not really sure what I was thinking when I wrote this. In this case, the measure is real valued.
real-analysis measure-theory
asked Dec 12 '18 at 7:06
James LengJames Leng
112
$endgroup$
add a comment |
$begingroup$
Let $B$ be a Banach space and $mu$ a $B$-valued measure on $X$. If $f: X to B$ is a $mu$-integrable function, then define
$$mu_f(E) = int_E f dmu$$
Is it true that the image of $mu_f$ is relatively compact in $B$? I guess the real question is: if $mu$ is totally finite measure, is it true that the image of $mu$ is relatively compact?
@below
You are right. Not really sure what I was thinking when I wrote this. In this case, the measure is real valued.
real-analysis measure-theory
asked Dec 12 '18 at 7:06
James LengJames Leng
112
$endgroup$
$begingroup$
Either $mu$ or $f$ has to take values in the field $mathbb{R}$ resp. $mathbb{C}$. (For your situation you need at least a Banach algebra.)
$endgroup$
– p4sch
Dec 12 '18 at 8:42
add a comment |
$begingroup$
Let $B$ be a Banach space and $mu$ a $B$-valued measure on $X$. If $f: X to B$ is a $mu$-integrable function, then define
$$mu_f(E) = int_E f dmu$$
Is it true that the image of $mu_f$ is relatively compact in $B$? I guess the real question is: if $mu$ is totally finite measure, is it true that the image of $mu$ is relatively compact?
@below
You are right. Not really sure what I was thinking when I wrote this. In this case, the measure is real valued.
real-analysis measure-theory
asked Dec 12 '18 at 7:06
James LengJames Leng
112
$endgroup$
Let $B$ be a Banach space and $mu$ a $B$-valued measure on $X$. If $f: X to B$ is a $mu$-integrable function, then define
$$mu_f(E) = int_E f dmu$$
Is it true that the image of $mu_f$ is relatively compact in $B$? I guess the real question is: if $mu$ is totally finite measure, is it true that the image of $mu$ is relatively compact?
@below
You are right. Not really sure what I was thinking when I wrote this. In this case, the measure is real valued.
real-analysis measure-theory
real-analysis measure-theory
asked Dec 12 '18 at 7:06
James LengJames Leng
112
asked Dec 12 '18 at 7:06
James LengJames Leng
112
edited Dec 12 '18 at 15:05
James Leng
asked Dec 12 '18 at 7:06
James LengJames Leng
112
asked Dec 12 '18 at 7:06
James LengJames Leng
112
asked Dec 12 '18 at 7:06
James LengJames Leng
112
112
$begingroup$
Either $mu$ or $f$ has to take values in the field $mathbb{R}$ resp. $mathbb{C}$. (For your situation you need at least a Banach algebra.)
$endgroup$
– p4sch
Dec 12 '18 at 8:42
add a comment |
$begingroup$
Either $mu$ or $f$ has to take values in the field $mathbb{R}$ resp. $mathbb{C}$. (For your situation you need at least a Banach algebra.)
$endgroup$
– p4sch
Dec 12 '18 at 8:42
$begingroup$
Either $mu$ or $f$ has to take values in the field $mathbb{R}$ resp. $mathbb{C}$. (For your situation you need at least a Banach algebra.)
$endgroup$
– p4sch
Dec 12 '18 at 8:42
$begingroup$
Either $mu$ or $f$ has to take values in the field $mathbb{R}$ resp. $mathbb{C}$. (For your situation you need at least a Banach algebra.)
$endgroup$
– p4sch
Dec 12 '18 at 8:42
add a comment |
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$begingroup$
Either $mu$ or $f$ has to take values in the field $mathbb{R}$ resp. $mathbb{C}$. (For your situation you need at least a Banach algebra.)
$endgroup$
– p4sch
Dec 12 '18 at 8:42