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Let $(Y, \Sigma,\mu)$ be measure space and $X$ a Polish space endowed with its Borel $\sigma$-algebra. Suppose that $f:Y\times X\to \mathbb R$ is a Carathéodory function (i.e. coninuous in $x\in X$ for each $y\in Y$, measurable and bounded by a $L^1$ function that does not depend on $x$). Let ${\Sigma}_0$ be sub $\sigma$-algebra of $\Sigma$, and let $g(\cdot,x)=E(f(\cdot,x)|\Sigma_0)$ denote the conditional expectation with respect to ${\Sigma}_0$. Does $g$ have a version that is a Carathéodory function as well?

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Let $(Y, \Sigma,\mu)$ be measure space and $X$ a Polish space endowed with its Borel $\sigma$-algebra. Suppose that $f:Y\times X\to \mathbb R$ is a Carathéodory function (i.e. continuous in $x\in X$ for each $y\in Y$, measurable and bounded by a $L^1$ function that does not depend on $x$). Let ${\Sigma}_0$ be sub $\sigma$-algebra of $\Sigma$, and let $g(\cdot,x)=E(f(\cdot,x)|\Sigma_0)$ denote the conditional expectation with respect to ${\Sigma}_0$. Does $g$ have a version that is a Carathéodory function as well?

Thanks!

Let $(Y,\mathcal Y)$ be measurable space and $X$ a Polish space endowed with its Borel $\sigma$-algebra. Suppose that $f:Y\times X\to \mathbb R$ is a Carathéodory function (i.e. coninuous in $x\in X$ for each $y\in Y$ and measurable). Let $\hat {\mathcal Y}$ be sub $\sigma$-algebra of $\mathcal Y$, and let $g(\cdot,x)=E(f(\cdot,x)|\hat{\mathcal Y})$ denote the conditional expectation with respect to $\hat{\mathcal Y}$. Is $g$ a Carathódory function as well?

Thanks!

Let $(Y, \Sigma,\mu)$ be measure space and $X$ a Polish space endowed with its Borel $\sigma$-algebra. Suppose that $f:Y\times X\to \mathbb R$ is a Carathéodory function (i.e. coninuous in $x\in X$ for each $y\in Y$, measurable and bounded by a $L^1$ function that does not depend on $x$). Let ${\Sigma}_0$ be sub $\sigma$-algebra of $\Sigma$, and let $g(\cdot,x)=E(f(\cdot,x)|\Sigma_0)$ denote the conditional expectation with respect to ${\Sigma}_0$. Does $g$ have a version that is a Carathéodory function as well?

Thanks!