Consider some measurable space $X$, for example $X = \mathbb{R}^n$, and two probability measures $p$ and $q$ on $X$. Do we generally have $L^2(p) \subseteq L^2(q)$? Under what conditions would the inclusion hold? Under what conditions would it be an equality?
Given some $f \in L^2(p)$, we know that
$$\|f\|_{L^2(p)}^2 := \int_X f(x)^2 \mathrm{d}p(x) < \infty. $$
Ideally, we'd like to show that:
$$\|f\|_{L^2(q)}^2 := \int_X f(x)^2 \mathrm{d}q(x) < \infty. $$
Using less rigourous notation, we can write:
$$\|f\|_{L^2(q)}^2 = \int_X f(x)^2 \frac{q(x)}{p(x)} p(x) \mathrm{d}x.$$
However, I don't know how to transform this last expression into something that involves $\|f\|_{L^2(p)}$, which is the only known quantity. I have tried using a measure change inequality, which lead to the inequality:
$$\|f\|_{L^2(q)}^2 \leq \log \left(\int_X e^{f(x)^2} p(x) \mathrm{d}x\right) + \mathrm{KL}(q||p).$$
However, there is no guarantee that the integral inside the logarithm is finite.
Is this a known question?