Vtyjkyuk

Theorem 10.1.1 in Prekopa establishes the following result.

Suppose $h:mathbbR^n times mathbbR^m to mathbbR$ is continuous and $xi$ is a random vector from a probability space $(Xi,mathcalF,mathbbP)$ with $Xi subset mathbbR^m$ and $mathbbPh(x,xi) = 0 = 0$, $forall x in mathbbR^n$. Let $mathcali$ denote the indicator function of the positive reals, i.e. $$mathcali(z) := begincases 1 &mboxif z > 0 \
0 & mboxif z leq 0endcases.$$

Then $mathbbEleft[mathcali(h(x,xi))right]$ is a continuous function of $x$, where $mathbbE$ denotes the expectation operator.

My question is the following: Let $g:mathbbR^n times mathbbR^m to mathbbR$ be a continuous function such that $leftlvert g(x,xi) rightrvert < M$ for some $M > 0$ (this ensures that the next expectation is well-defined). Do the above conditions also guarantee that $mathbbEleft[leftlvertmathcali(h(x,xi)) - g(x,xi)rightrvertright]$ is a continuous function of $x$, or do we need to impose additional assumptions?

Clearly, we have that the above expectation is continuous if the absolute value is omitted. I have a feeling that continuity still holds even if we include the absolute value. I've tried to use dominated convergence to show this - in the process, I've had to show that for each sequence $x_n to x$ and $xi in Xi$, $undersetn to inftylim mathcali(h(x_n,xi))$ converges to $mathcali(h(x,xi))$ in the $L^1$-norm, which eludes me.