Consider a probability space $(\Omega,\mathcal H,P)$ and a real random variable $X$ such that $E(X)$ is well defined (also infinite values are allowed). Is it true that the measure $$\mu(A):=E(X\chi_A)=\int_A X\,dP$$ is a $\sigma$-finite measure? Clearly if $X\in L^1$ we obtain a finite measure, but I'm interested at the general case when $E(X)$ can be also $\infty$.
Thans in advance.
By "real random variable" you mean the values are real? That is, we have $P(X=\infty)=0$ ? Then sets $$ A_n = \{ X < n \} $$ have finite measure, and $$ \Omega = \bigcup_{n} A_n . $$