# Can we interchange integral and limit if the sequence of function is uniformly integrable?

Assumptions:

1) $$h(x,\omega)$$ is defined on $$\forall\omega\in\Omega, \forall x\in R\setminus\{x_0\}$$ where $$\omega$$ is a random variable.

2) Let $$h(x_0,\omega)=\lim\limits_{x\rightarrow x_0}h(x,\omega)$$ (pointwise convergence).

3) $$h(x,\omega)$$ is integrable $$\forall x\in R$$.

4) $$\lim\limits_{x\rightarrow x_0}\int_\Omega h(x,\omega)dP(\omega)$$ exists.

5) The family $$h(x,\omega)$$ of random variables for all $$x$$ sufficiently close to $$x_0$$ is uniformly integrable. That is to say, for each $$\epsilon>0$$, there exists $$C(\epsilon)$$ such that $$\limsup\limits_{x\rightarrow x_0}\int_{\{|h(x,\omega)|>C(\epsilon)\}}|h(x,\omega)|dP(\omega)<\epsilon$$.

Now is this true?

$$\int_\Omega h(x_0,\omega)dP(\omega)=\lim\limits_{x\rightarrow x_0}\int_\Omega h(x,\omega)dP(\omega)$$

I can't see how to apply Lebesgue dominated convergence theorem here because it's hard to find a function to dominate all $$h(x,\omega)$$. Any ideas will be appreciated.

It suffices to show that for all sequence $$\left(x_n\right)_{n\geqslant 1}$$ converging to $$x_0$$, the convergence $$\int_\Omega h(x_0,\omega)dP(\omega)=\lim\limits_{n\rightarrow +\infty}\int_\Omega h(x_n,\omega)dP(\omega).$$ Let us define the random variables $$Y_n\colon \omega\mapsto h(x_n,\omega)$$, $$n\geqslant 0$$. Item 6 of this answer applies.