If ${\mathbb E} [f(x)] < \infty$, does $\|f(x)p(x)\|_{q}$ hold for $q \in [1, \infty]$? Note: $p(x)$ is the pdf of $X$.

Question

For map $f: {\mathcal X} \to [0, \infty)$ (where ${\mathcal X} \subseteq {\mathbb R}^n$), let

$${\mathbb E} [f(x)] = \int_{{\mathcal X}} f(x) p(x) dx < \infty,$$

where $p(x)$ is the pdf of $X$.

Then, does the following hold?

$\|f(x)p(x)\|_{q} = \left\{\int_{{\mathcal X}} [f(x) p(x)]^q dx\right\}^{1/q} < \infty,$

where $q \in [1, \infty]$.

If it doesn't, can we find some counterexamples?

Thanks!

Answer 1

Let $q\in (1,\infty)$ and $p\in L^{1}\backslash L^{q}$. Then $f=1$ and $p/\|p\|_{1}$ are a counterexample.