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relevant subject tag+ abbreviation is little known, so I "expanded" it
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edit approved Oct 13, 2018 at 16:43
Amir Sagiv
edit approved Oct 13, 2018 at 16:43
Amir Sagiv
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Setup

Suppose that $U$ is a subset of $L^{\infty}_{\mu}(\mathscr{F})\cap L^1_{\mu}(\mathscr{F})$ defined by $$ f\in U \Leftrightarrow g(f(x))\leq M \mbox{ and } f \in L^{\infty}_{\mu}(\mathscr{F})\cap L^1_{\mu}(\mathscr{F}) , $$ for a fixed convex and l.s.c.lower-semicontinuous function $g$ . Here $\mu$ is a (if acceptable $\sigma$-)finite measure on $\mathcal{X}$.

Question

Under what conditions can the interchanging of the $$ \inf_{f \in U}\int_{x \in \mathcal{X}}f^2(x)\mu(dx) = \int_{x \in \mathcal{X}}\inf_{f \in U}f^2(x)\mu(dx). $$ In other words....when can the infimum be taken pointwise?

Setup

Suppose that $U$ is a subset of $L^{\infty}_{\mu}(\mathscr{F})\cap L^1_{\mu}(\mathscr{F})$ defined by $$ f\in U \Leftrightarrow g(f(x))\leq M \mbox{ and } f \in L^{\infty}_{\mu}(\mathscr{F})\cap L^1_{\mu}(\mathscr{F}) , $$ for a fixed convex and l.s.c. function $g$ . Here $\mu$ is a (if acceptable $\sigma$-)finite measure on $\mathcal{X}$.

Question

Under what conditions can the interchanging of the $$ \inf_{f \in U}\int_{x \in \mathcal{X}}f^2(x)\mu(dx) = \int_{x \in \mathcal{X}}\inf_{f \in U}f^2(x)\mu(dx). $$ In other words....when can the infimum be taken pointwise?

Setup

Suppose that $U$ is a subset of $L^{\infty}_{\mu}(\mathscr{F})\cap L^1_{\mu}(\mathscr{F})$ defined by $$ f\in U \Leftrightarrow g(f(x))\leq M \mbox{ and } f \in L^{\infty}_{\mu}(\mathscr{F})\cap L^1_{\mu}(\mathscr{F}) , $$ for a fixed convex and lower-semicontinuous function $g$ . Here $\mu$ is a (if acceptable $\sigma$-)finite measure on $\mathcal{X}$.

Question

Under what conditions can the interchanging of the $$ \inf_{f \in U}\int_{x \in \mathcal{X}}f^2(x)\mu(dx) = \int_{x \in \mathcal{X}}\inf_{f \in U}f^2(x)\mu(dx). $$ In other words....when can the infimum be taken pointwise?

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ABIM
ABIM
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Echange of Infimum Integral with Pointwise Infimum

Setup

Suppose that $U$ is a subset of $L^{\infty}_{\mu}(\mathscr{F})\cap L^1_{\mu}(\mathscr{F})$ defined by $$ f\in U \Leftrightarrow g(f(x))\leq M \mbox{ and } f \in L^{\infty}_{\mu}(\mathscr{F})\cap L^1_{\mu}(\mathscr{F}) , $$ for a fixed convex and l.s.c. function $g$ . Here $\mu$ is a (if acceptable $\sigma$-)finite measure on $\mathcal{X}$.

Question

Under what conditions can the interchanging of the $$ \inf_{f \in U}\int_{x \in \mathcal{X}}f^2(x)\mu(dx) = \int_{x \in \mathcal{X}}\inf_{f \in U}f^2(x)\mu(dx). $$ In other words....when can the infimum be taken pointwise?