Suppose $X,Y$ are real-valued random vectors. When can we say $E[g(X,Y)|X]$ is an a.e. continuous function in $X$? I ask because I am reading Chapter 7 of Hansen, which asserts without proof that the finite-sample conditional variance of the OLS estimator converges in probability to the asymptotic variance of the OLS estimator (p.167). I have mostly convinced myself of this argument, save for one step where I want to argue $E\left[\frac{1}{n}\sum_{i=1}^{n}X_{i}X_{i}'e_{i}^{2}|(X_1,...,X_n)'\right]\overset{p}{\rightarrow}E\left[E\left[X_{i}X_{i}'e_{i}^{2}\right]|(X_1,...,X_n)'\right]=E\left[X_{i}X_{i}'e_{i}^{2}\right]$ by appealing to the weak law of large numbers and the continuous mapping theorem.
This post seems to suggest "near continuity," but not sure what precisely that means or if that would suffice here. Any help is much appreciated!
The error is $e_i=Y_i-X_i'\beta$ and model assumptions include $(X_i,Y_i)$ iid and finite fourth moments of $X_i,Y_i$. Then the claim can be shown without appealing to the continuous mapping theorem since
$$E\left[\frac{1}{n}\sum_{i=1}^{n}X_{i}X_{i}'e_{i}^{2}|(X_1,...,X_n)'\right]= \frac{1}{n}\sum_{i=1}^{n}X_{i}X_{i}'E\left[e_{i}^{2}|(X_1,...,X_n)'\right]\\ =\frac{1}{n}\sum_{i=1}^{n}X_{i}X_{i}'E\left[e_{i}^{2}|X_i\right]\quad ((X_i,Y_i) \text{ iid})\\ \overset{p}{\rightarrow}E\left[X_{i}X_{i}'E\left[e_{i}^{2}|X_i\right]\right]\quad (\text{WLLN})\\ =E\left[E\left[X_{i}X_{i}'e_{i}^{2}|X_i\right]\right]\\= E\left[X_{i}X_{i}'e_{i}^{2}\right]\quad (\text{LIE})$$