Say $X_1,\ldots,X_n$ is a random sample from a multivariate normal distribution $N(\boldsymbol{\beta},\Sigma)$ where $\Sigma$ is known, and $\widehat{\boldsymbol{\beta}}_n = (1/n)\sum_{i=1}^n X_i$. Consider the following integral: $$ I(\widehat{\boldsymbol{\beta}}_n) := \int f(\boldsymbol{a}) \phi(\boldsymbol{a}\mid \widehat{\boldsymbol{\beta}}_n,\Sigma/n)\,\mathrm{d}\boldsymbol{a}, $$ where $\phi(x\mid \mu,\Sigma)$ is the density of the multivariate normal distribution with mean vector $\mu$ and covariance matrix $\Sigma$.
My claim/conjecture is that as long as $f$ is a compactly supported continuous function, $I(\widehat{\boldsymbol{\beta}}_n) \xrightarrow{p} f(\boldsymbol{\beta})$, implying that $\phi(\boldsymbol{a}\mid \widehat{\boldsymbol{\beta}}_n,\Sigma/n) \to \delta(\boldsymbol{a}-\boldsymbol{\beta})$ in some mode of convergence. To prove this, I need to replace $\widehat{\boldsymbol{\beta}}_n$ with $\boldsymbol{\beta}$, but consistency doesn't help here. How should I go about proving or disproving this?