show $|int_X f dmu-int_B f dmu| <epsilon$ for all $Bin F$integral of a measure zero setUpper bound on the integral of a function given the measure of the integration domain$exists EinmathcalX$ such that $mu(E)<infty$ and $int_X|f|dmu<int_E|f|dmu + epsilon$relation between measure and integral for all $epsilon >0$$lim_n to infty int_X f_n , dmu = int_X f , dmu$ implies $lim_n to infty int_B f_n , dmu = int_B f , dmu$ for $B subseteq X$show $lim_nrightarrowinftyint_X|f_n-f|dmu=0$Prove $int_X |f|^p=pint^infty_0 t^p-1mu(f(x)>t) dt,$proving $|int_Xfdmu|leqint_X|f|dmu$?If $(f_j)to f$ in measure, show that $int_X f,dmulevarliminfint_X f_j,dmu$prove that $int_X f=int_[0,infty)mu(f(x)>t)dm(t)$
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show $|int_X f dmu-int_B f dmu|
integral of a measure zero setUpper bound on the integral of a function given the measure of the integration domain$exists EinmathcalX$ such that $mu(E)<infty$ and $int_X|f|dmu<int_E|f|dmu + epsilon$relation between measure and integral for all $epsilon >0$$lim_n to infty int_X f_n , dmu = int_X f , dmu$ implies $lim_n to infty int_B f_n , dmu = int_B f , dmu$ for $B subseteq X$show $lim_nrightarrowinftyint_X|f_n-f|dmu=0$Prove $int_X |f|^p=pint^infty_0 t^p-1mu(f(x)>t) dt,$proving $|int_Xfdmu|leqint_X|f|dmu$?If $(f_j)to f$ in measure, show that $int_X f,dmulevarliminfint_X f_j,dmu$prove that $int_X f=int_[0,infty)mu(x)dm(t)$
$begingroup$
I have following task, that I just do not know how to solve and especially I do not know how this A can help here.
Let $(X,F,mu)$ be a measure space $f:Xrightarrow mathbbR$ integrable. $A,Bin F$ with $Asubset B$ and $mu(A)<infty$. Show that $|int_X f dmu-int_B f dmu|<epsilon$ $forall$ $Bin F$.
measure-theory
asked Mar 29 at 20:13
tim123tim123
195
$endgroup$
add a comment |
$begingroup$
I have following task, that I just do not know how to solve and especially I do not know how this A can help here.
Let $(X,F,mu)$ be a measure space $f:Xrightarrow mathbbR$ integrable. $A,Bin F$ with $Asubset B$ and $mu(A)<infty$. Show that $|int_X f dmu-int_B f dmu|<epsilon$ $forall$ $Bin F$.
measure-theory
asked Mar 29 at 20:13
tim123tim123
195
$endgroup$
1
$begingroup$
You are missing some hypothesis: take $X=mathbbR$ with the usual Lebesgue measure, $A = emptyset$ and $B$ such that $mu(B)=0$, then you want to prove that $| int_mathbbR f dmu | < epsilon$ for all integrable functions...
$endgroup$
– dcolazin
Mar 29 at 21:07
add a comment |
$begingroup$
I have following task, that I just do not know how to solve and especially I do not know how this A can help here.
Let $(X,F,mu)$ be a measure space $f:Xrightarrow mathbbR$ integrable. $A,Bin F$ with $Asubset B$ and $mu(A)<infty$. Show that $|int_X f dmu-int_B f dmu|<epsilon$ $forall$ $Bin F$.
measure-theory
asked Mar 29 at 20:13
tim123tim123
195
$endgroup$
I have following task, that I just do not know how to solve and especially I do not know how this A can help here.
Let $(X,F,mu)$ be a measure space $f:Xrightarrow mathbbR$ integrable. $A,Bin F$ with $Asubset B$ and $mu(A)<infty$. Show that $|int_X f dmu-int_B f dmu|<epsilon$ $forall$ $Bin F$.
measure-theory
measure-theory
asked Mar 29 at 20:13
tim123tim123
195
asked Mar 29 at 20:13
tim123tim123
195
edited Mar 29 at 20:20
tim123
asked Mar 29 at 20:13
tim123tim123
195
asked Mar 29 at 20:13
tim123tim123
195
asked Mar 29 at 20:13
tim123tim123
195
195
1
$begingroup$
You are missing some hypothesis: take $X=mathbbR$ with the usual Lebesgue measure, $A = emptyset$ and $B$ such that $mu(B)=0$, then you want to prove that $| int_mathbbR f dmu | < epsilon$ for all integrable functions...
$endgroup$
– dcolazin
Mar 29 at 21:07
add a comment |
1
$begingroup$
You are missing some hypothesis: take $X=mathbbR$ with the usual Lebesgue measure, $A = emptyset$ and $B$ such that $mu(B)=0$, then you want to prove that $| int_mathbbR f dmu | < epsilon$ for all integrable functions...
$endgroup$
– dcolazin
Mar 29 at 21:07
1
1
$begingroup$
You are missing some hypothesis: take $X=mathbbR$ with the usual Lebesgue measure, $A = emptyset$ and $B$ such that $mu(B)=0$, then you want to prove that $| int_mathbbR f dmu | < epsilon$ for all integrable functions...
$endgroup$
– dcolazin
Mar 29 at 21:07
$begingroup$
You are missing some hypothesis: take $X=mathbbR$ with the usual Lebesgue measure, $A = emptyset$ and $B$ such that $mu(B)=0$, then you want to prove that $| int_mathbbR f dmu | < epsilon$ for all integrable functions...
$endgroup$
– dcolazin
Mar 29 at 21:07
add a comment |
0
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$begingroup$
You are missing some hypothesis: take $X=mathbbR$ with the usual Lebesgue measure, $A = emptyset$ and $B$ such that $mu(B)=0$, then you want to prove that $| int_mathbbR f dmu | < epsilon$ for all integrable functions...
$endgroup$
– dcolazin
Mar 29 at 21:07