I am trying to figure out if there are some particular mathematical properties that PDF-s for continuous distributions follow.
Let's assume that we have some continuous random variable $X$ that is distributed according to PDF $p_x$. Let's also assume that mathematical expectation of the variable is finite. (1) Will it mean that $p_x$ does not have infinity values anywhere, meaning that no set of $x$ that has non-zero Lebesgue measure turns $p_x$ to infinity? (2) Will it also mean then that any function of $X$ will have a finite mathematical expectation, except for the functions that turn to infinity on the set of $x$-s that has non-zero measure under $p_x$-induced measure?