I've been working on a problem and at some point in the proof I need to show that the following set $$\left\{\mu \in \mathcal{P}_{ac}(\mathbb R^d): \int \varphi(x)\mu(\mathrm{d}x)\leq C\right\}$$ is a compact with respect to the topology induced by TVtotal-variation distance on $\mathcal{P}_{ac}(\mathbb R^d).$ Here, $\mathcal{P}_{ac}(\mathbb R^d)$ is the set of probability measures absolutely continuous with respect to Lebesgue measure on $\mathbb R^d$, $\varphi$ is a real-valued bounded measurable function on $\mathbb R^d$ and $C > 0$ is a constant.
What I've thought about so far: Prokhorov's theorem is not helpful here since it only gives compactness with respect to the weak convergence of measures with is a weaker topology than the one induced by TVtotal-variation distance. I know that for Wasserstein spaces there is this result: for any $p'> p \geq 1,$ the set $$\left\{\mu \in \mathcal{P}_{p}(\mathbb R^d): \int |x|^p\mu(\mathrm{d}x)\leq C\right\}$$ is $W_p$-compact, $W_p$ is known to metrize weak convergence on $\mathcal{P}_{p}(\mathbb R^d).$
I would greatly appreciate if someone could point out a reference for what I'm trying to prove or to give me some ideas how to prove it. Thank you!
