Compactness of the space of measure-preserving maps
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Let $X, Y$ be Polish spaces and $mu, nu$ be Borel probablity measures on $X, Y$, respectively. Assume that $mu$ is non-atomic. Denote by $mathcal{M}(mu,nu)$ a set of Borel measurable mappings $T$ from $X$ to $Y$ which preserve measures (i.e. for any Borel set $B$ of $Y$, we have $mu(T^{-1}(B))=nu(B)$). Is the set $mathcal{M}(mu,nu)$ compact with respect to the convergence of $mu$-measure?
real-analysis measure-theory
asked Dec 7 '18 at 11:22
NickNick
162
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add a comment |
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Let $X, Y$ be Polish spaces and $mu, nu$ be Borel probablity measures on $X, Y$, respectively. Assume that $mu$ is non-atomic. Denote by $mathcal{M}(mu,nu)$ a set of Borel measurable mappings $T$ from $X$ to $Y$ which preserve measures (i.e. for any Borel set $B$ of $Y$, we have $mu(T^{-1}(B))=nu(B)$). Is the set $mathcal{M}(mu,nu)$ compact with respect to the convergence of $mu$-measure?
real-analysis measure-theory
asked Dec 7 '18 at 11:22
NickNick
162
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What do mean with 'with respect to the convergence of $mu$-measure'?
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– p4sch
Dec 7 '18 at 12:29
1
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I made some mistakes. Convergence ''in'' $mu$-measure, not convergence ''of'' $mu$-measure. We need to fix a distance $d$ on $Y$. A sequence ${T_n}$ of Borel measurable maps from $X$ to $Y$ converges in $mu$-measure to a Borel measurable map $Tcolon Xto Y$ if for any $epsilon>0$, $mu({xin X mid d(T_n(x),T(x))>varepsilon})$ tends to $0$ as $n$ tends to infinity.
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– Nick
Dec 7 '18 at 13:04
add a comment |
$begingroup$
Let $X, Y$ be Polish spaces and $mu, nu$ be Borel probablity measures on $X, Y$, respectively. Assume that $mu$ is non-atomic. Denote by $mathcal{M}(mu,nu)$ a set of Borel measurable mappings $T$ from $X$ to $Y$ which preserve measures (i.e. for any Borel set $B$ of $Y$, we have $mu(T^{-1}(B))=nu(B)$). Is the set $mathcal{M}(mu,nu)$ compact with respect to the convergence of $mu$-measure?
real-analysis measure-theory
asked Dec 7 '18 at 11:22
NickNick
162
$endgroup$
Let $X, Y$ be Polish spaces and $mu, nu$ be Borel probablity measures on $X, Y$, respectively. Assume that $mu$ is non-atomic. Denote by $mathcal{M}(mu,nu)$ a set of Borel measurable mappings $T$ from $X$ to $Y$ which preserve measures (i.e. for any Borel set $B$ of $Y$, we have $mu(T^{-1}(B))=nu(B)$). Is the set $mathcal{M}(mu,nu)$ compact with respect to the convergence of $mu$-measure?
real-analysis measure-theory
real-analysis measure-theory
asked Dec 7 '18 at 11:22
NickNick
162
asked Dec 7 '18 at 11:22
NickNick
162
asked Dec 7 '18 at 11:22
NickNick
162
asked Dec 7 '18 at 11:22
NickNick
162
asked Dec 7 '18 at 11:22
NickNick
162
162
$begingroup$
What do mean with 'with respect to the convergence of $mu$-measure'?
$endgroup$
– p4sch
Dec 7 '18 at 12:29
1
$begingroup$
I made some mistakes. Convergence ''in'' $mu$-measure, not convergence ''of'' $mu$-measure. We need to fix a distance $d$ on $Y$. A sequence ${T_n}$ of Borel measurable maps from $X$ to $Y$ converges in $mu$-measure to a Borel measurable map $Tcolon Xto Y$ if for any $epsilon>0$, $mu({xin X mid d(T_n(x),T(x))>varepsilon})$ tends to $0$ as $n$ tends to infinity.
$endgroup$
– Nick
Dec 7 '18 at 13:04
add a comment |
$begingroup$
What do mean with 'with respect to the convergence of $mu$-measure'?
$endgroup$
– p4sch
Dec 7 '18 at 12:29
1
$begingroup$
I made some mistakes. Convergence ''in'' $mu$-measure, not convergence ''of'' $mu$-measure. We need to fix a distance $d$ on $Y$. A sequence ${T_n}$ of Borel measurable maps from $X$ to $Y$ converges in $mu$-measure to a Borel measurable map $Tcolon Xto Y$ if for any $epsilon>0$, $mu({xin X mid d(T_n(x),T(x))>varepsilon})$ tends to $0$ as $n$ tends to infinity.
$endgroup$
– Nick
Dec 7 '18 at 13:04
$begingroup$
What do mean with 'with respect to the convergence of $mu$-measure'?
$endgroup$
– p4sch
Dec 7 '18 at 12:29
$begingroup$
What do mean with 'with respect to the convergence of $mu$-measure'?
$endgroup$
– p4sch
Dec 7 '18 at 12:29
1
1
$begingroup$
I made some mistakes. Convergence ''in'' $mu$-measure, not convergence ''of'' $mu$-measure. We need to fix a distance $d$ on $Y$. A sequence ${T_n}$ of Borel measurable maps from $X$ to $Y$ converges in $mu$-measure to a Borel measurable map $Tcolon Xto Y$ if for any $epsilon>0$, $mu({xin X mid d(T_n(x),T(x))>varepsilon})$ tends to $0$ as $n$ tends to infinity.
$endgroup$
– Nick
Dec 7 '18 at 13:04
$begingroup$
I made some mistakes. Convergence ''in'' $mu$-measure, not convergence ''of'' $mu$-measure. We need to fix a distance $d$ on $Y$. A sequence ${T_n}$ of Borel measurable maps from $X$ to $Y$ converges in $mu$-measure to a Borel measurable map $Tcolon Xto Y$ if for any $epsilon>0$, $mu({xin X mid d(T_n(x),T(x))>varepsilon})$ tends to $0$ as $n$ tends to infinity.
$endgroup$
– Nick
Dec 7 '18 at 13:04
add a comment |
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$begingroup$
What do mean with 'with respect to the convergence of $mu$-measure'?
$endgroup$
– p4sch
Dec 7 '18 at 12:29
1
$begingroup$
I made some mistakes. Convergence ''in'' $mu$-measure, not convergence ''of'' $mu$-measure. We need to fix a distance $d$ on $Y$. A sequence ${T_n}$ of Borel measurable maps from $X$ to $Y$ converges in $mu$-measure to a Borel measurable map $Tcolon Xto Y$ if for any $epsilon>0$, $mu({xin X mid d(T_n(x),T(x))>varepsilon})$ tends to $0$ as $n$ tends to infinity.
$endgroup$
– Nick
Dec 7 '18 at 13:04