Suppose we have a parametric continuous function of $x\in\mathbb{R}$ with d-dimensional parameter $\theta$
$$g(x;\theta)$$
we also have have an n-dimensional sample if i.i.d. observations of X.
With this sample we compute:

1. a "generic" statistic $\hat{\theta}\_{n}$ (not necessarily an MLE) convergent in probability to a finite value that I plug in place of $\theta$;
2. a kernel density function $\hat{f}\_{n,h}(x)$ with smoothing parameter h.

Suppose we want to compute an estimate of the expectation of $g$ as the following Integral $I(\hat{\theta}\_{n},\hat{f}\_{n,h})$

$$I(\hat{\theta}\_{n},\hat{f}\_{n,h}):=\int\_{-\infty}^{\infty} g(x;\hat{\theta}\_{n})\hat{f}\_{n,h}(x)dx$$

My question is:

Am I allowed to state that for the continuous mapping theorem <https://en.wikipedia.org/wiki/Continuous_mapping_theorem> the integral $I(\hat{\theta}\_{n},\hat{f}\_{n,h})$ converges in probability?

Am I allowed to study the convergence as a Statistical functional assuming the Hadamard differentiability of $I(,)$? How to do it in this case?

thank you!