Proof involving weak convergence: where to us compactness I have the following claim to prove as homework:
Consider a continuous random variable $W$ with PDF $f_W(\cdot)$ and probability measure $\mathbb{P}_W$. Let $B$ be a compact  subset of $\mathbb{R}\times \mathbb{R}^+$. Show that, if $B$ is large enough, then there exists a probability measure  $\pi(\cdot)$ on $B$
such that $f_W(\cdot)$. can be approximated as
\begin{equation}
\label{main}
f_W(w)\approx \int_{B} g(w; \mu, \sigma^2) d\pi(\mu, \sigma^2) \quad \text{ for each $w\in \mathbb{R}$},
\end{equation}
where  $g(\cdot; \mu, \sigma^2)$ is the PDF of a Normally distributed r.v. with mean $\mu$ and variance $\sigma^2$.
I have written a proof (reported below). However, there are 2 things which I do not understand and I would like your help on:
(1) Where should I use compactness of $B$ (if needed at all)?
(2) Suppose the $f_W(\cdot)$ has very fat tails. Then, I expect the approximation to be worse than for another PDF with thin tails. Where does this show up in my derivations?

My proof:
Step 1: Let $\mathbb{P}_W(\cdot)$ be the probability measure associated with $f_W(\cdot)$. Consider a sequence of  finite and strictly positive numbers $\{s^2_n\}_n$ such that $\lim_{n\rightarrow \infty}s^2_n=0$. Consider a sequence of r.v. $\{Y_n\}_n$ such that each $Y_n\sim \mathcal{N}(0,s^2_n)$ and $Y_n\perp W$. Let $Z_n\equiv W+Y_n$. Then, by Slutsky's Theorem, it holds that:
$$
 Z_n\rightarrow_d W \quad \text{as $n\rightarrow \infty$},
 $$
that is:
$$
(1) \quad \lim_{n\rightarrow \infty} \int_{x\in \mathbb{R}} h (x) d \mathbb{P}_{Z_n}(x) = \int_{x\in \mathbb{R}} h (x) d \mathbb{P}_{W}(x) \quad \text{for any bounded continuous function $h:\mathbb{R}\rightarrow \mathbb{R}$},
$$
where $\mathbb{P}_{Z_n}(\cdot)$ is the probability measure associated with the PDF $f_{Z_n}(\cdot)$ of $Z_n$.
Note also that
$$
f_{Z_n}(z)=\int_{\mu \in \mathbb{R}} g(z; \mu, s^2_n) d \mathbb{P}_W(\mu) \quad \text{ for each $z\in \mathbb{R}$},
$$
where $g(\cdot; \mu, s^2_n)$ is the PDF of a Normally distributed r.v. with mean $\mu$ and variance $s^2_n$.
In turn, (1) can be rewritten as
$$
\lim_{n\rightarrow \infty}\int_{\mu \in \mathbb{R}} \Big( \int_{x\in \mathbb{R}} h (x)  d \mathbb{G}(x; \mu, s^2_n)\Big) d \mathbb{P}_{W}(\mu)= \int_{x\in \mathbb{R}} h (x) d \mathbb{P}_{W}(x)  \quad \text{for any bounded continuous function $h:\mathbb{R}\rightarrow \mathbb{R}$},
$$
where $\mathbb{G}(\cdot; \mu,s^2_n)$ is the probability measure associated with $g(\cdot; \mu, s^2_n)$.
Step 2:  Consider a sequence of finite and strictly positive numbers $\{\tau_n\}_n$ such that $\lim_{n\rightarrow \infty} \tau_n=0$. Consider a sequence of  sets $\{\mathcal{A}_{ n}\}_{n}\subset \mathbb{R}$ such that $\mathbb{P}_W(\mathcal{A}_{ n})>1-\tau_n$. Note that, as $\tau_n\rightarrow 0$, we have that the set $\mathcal{A}_{ n}$ enlarges. Let $\tilde{Z}_n$ be a r.v. with PDF given by the following Gaussian mixture:
$$
 f_{\tilde{Z}_n}(z)= \int_{\mu \in \mathcal{A}_{ n}}  g(z; \mu, s^2_n) d \mathbb{P}_W(\mu) \quad \text{for each $z\in \mathbb{R}$}.
 $$
In what follows we show that
$$
 \tilde{Z}_n\rightarrow_d W \quad \text{as $n\rightarrow \infty$}.
 $$
Step 3: Consider  any bounded continuous function $h:\mathbb{R}\rightarrow \mathbb{R}$ and let $M\equiv \sup_{x\in \mathbb{R}}|h(x)|<\infty$.   Then,
$$
 \Big| \int_{\mu \in \mathbb{R}} \Big( \int_{x\in \mathbb{R}} h (x)  d \mathbb{G}(x; \mu, s^2_n)\Big) d \mathbb{P}_{W}(\mu)  - \int_{\mu\in \mathcal{A}_{ n}}  \Big(\int_{x\in \mathbb{R}} h (x)  d \mathbb{G}(x; \mu, s^2_n)\Big) d \mathbb{P}_{W}(\mu)  \Big|\\
\leq    \int_{\mu \in \mathbb{R}\setminus \mathcal{A}_{ n}}  \Big( \int_{x\in \mathbb{R}} |h (x)|  d \mathbb{G}(x; \mu, s^2_n)\Big) d \mathbb{P}_{W}(\mu)\\
\leq   M\times \mathbb{P}_W(\mathbb{R}\setminus \mathcal{A}_{ n})\leq M\times \tau_n. 
$$
Further,
$$
  \lim_{n\rightarrow \infty} M\times \tau_n=0.
$$
Therefore:
$$
\lim_{n\rightarrow \infty}  \int_{\mu \in \mathcal{A}_{ n}}  \Big(\int_{x\in \mathbb{R}} h (x)  d \mathbb{G}(x; \mu, s^2_n)\Big) d \mathbb{P}_{W}(\mu) =\int_{x\in \mathbb{R}} h (x) d \mathbb{P}_{W}(x) \quad \text{for any bounded continuous function $h:\mathbb{R}\rightarrow \mathbb{R}$}. 
$$
By Fubini's Theorem,
$$
\int_{\mu \in \mathcal{A}_{ n}}  \Big(\int_{x\in \mathbb{R}} h (x)  d \mathbb{G}(x; \mu, s^2_n)\Big) d \mathbb{P}_{W}(\mu) = \int_{x\in \mathbb{R}} h(x) d\mathbb{P}_{\tilde{Z}_n}(x). 
 $$
Therefore:
$$
\lim_{n\rightarrow \infty} \int_{x\in \mathbb{R}} h(x) d\mathbb{P}_{\tilde{Z}_n}(x) =\int_{x\in \mathbb{R}} h (x) d \mathbb{P}_{W}(x) \quad \text{for any bounded continuous function $h:\mathbb{R}\rightarrow \mathbb{R}$},
$$
that is
$$
 \tilde{Z}_n \rightarrow_d W \text{ as $n\rightarrow \infty$}. 
 $$
Step 4: Let the set $\mathcal{B}_n$ be large enough to contain $\mathcal{A}_{ n}\times (0,\epsilon_n)$. We have that
$$
f_{\tilde{Z}_n}(z)= \int_{\mu \in \mathcal{A}_{ n}}  g(z; \mu, s^2_n) d \mathbb{P}_W(\mu)=\int_{\mu \in B_{ n}}  g(z; \mu, s^2_n) d \underbrace{(\mathbb{P}_W(\mu)\times \ {1}\{\mu\in \mathcal{A}_{n}, \sigma^2=s^2_n\})}_{\equiv \pi(\mu, \sigma^2)} \quad \text{for each $z\in \mathbb{R}$}.
$$
Therefore, for large $n$, we can approximate $f_W(\cdot)$ with $f_{\tilde{Z}_n}(\cdot)$ provided that $\mathcal{B}_n$ is large enough.
 A: For Question (1):
This compactness is useless. In fact, the question is equivalent to this one:

Let $B$ be a subset of $\mathbb{R}\times \mathbb{R}^+$. Show that, if $B$ is large enough, then there exists a probability measure  $\pi(\cdot)$ on $B$
such that $f_W(\cdot)$. can be approximated as
$$
f_W(w)\approx \int_{B} g(w; \mu, \sigma^2) d\pi(\mu, \sigma^2) \quad \text{ for each $w\in \mathbb{R}$},
$$
where  $g(\cdot; \mu, \sigma^2)$ is the PDF of a Normally distributed r.v. with mean $\mu$ and variance $\sigma^2$.

To see the equivalence, you can always properly choose $\pi(\mu,\sigma^2)$ so that its support is compact.
For Question (2):
When $f$ has a fat tail, you should choose larger $B$ to make sure the approximation is close. This means the approximation is slow.
To see this, in your step 3, since you choose $\mathcal{A}_{ n}$ such that $$\tag{*}\mathbb{P}_W(\mathbb{R}\setminus \mathcal{A}_{ n})>1-\tau_n\,.$$ When $f$ has a fat tail, you need to choose a larger $A_n$ to guarantee (*). Combining this with your step (4), this gives a larger $B$.
