I am interested to know which measurable functions are integrable with respect to all probability measures, i.e., all $f$, for which: $$ \int \| f \| ~{\rm d}\mathbb{P} < \infty,$$ where $\mathbb{P}$ is any arbitrary real valued probability measure.
I have come across this topic: Characterize functions that are integrable with respect to all probability measures
which says that all bounded measurable functions on $\mathbb{R}$ are integrable w.r.t. to any probability measure. So I assume this means all functions of the form $f:\mathbb{R}^n \rightarrow \mathbb{R}$. Is this correct? If so can anyone give me a proof why this is the case? (Or point me to a book with a proof)
It is easy to see that a bounded and measurable function $f:\mathbb R\to\mathbb R$ is integrable with respect to any probability measure $\mathbb P$ on the real line. If $\mathbb P$ is such a measure, and we let $K=\sup_{x\in\mathbb R}\lvert f(x)\rvert<\infty$, then $$ \int \lvert f\rvert\,\mathrm d\mathbb P\leq\int K\,\mathrm d\mathbb P=K\cdot\mathbb P(\mathbb R)=K<\infty. $$