If for all $\epsilon>0$, there exists $\delta>0$ such that $E[1_A|X|]<\epsilon$ for all measurable $P(A)<\delta$, then $E|X|<\infty$

Question

Let $X$ be a real valued random variable

If for all $\epsilon>0$, there exists $\delta>0$ such that $E[1_A|X|]<\epsilon$ for all measurable $A$ with $P(A)<\delta$, then $E|X|<\infty$.

I have no clue where to start.

Answer 1

Hint: you're assuming $X$ is real-valued.

As a result: given $\epsilon>0$, there exists $n=n(\epsilon)$ such that $P(|X|>n) <\delta(\epsilon)$.

Observe that each of the events $A_n = \{|X|>n\}$ decrease to $\emptyset$. Therefore by continuity of probability measures for monotone sequences of events you have $P(A_n) \to 0$. Fix any $\epsilon=1$ (or any $\epsilon>0$ for that matter). Let $n$ large enough so that $P(A_n)<\delta(\epsilon)$. Note that on $A_n^c$, $|X|\le n$. Then $$E[|X|] = E[|X|{\bf 1}_{A_n} ] + E[|X|{\bf 1}_{A_n^c}]<1 + n P(A_n^c) \le 1 + n<\infty.$$

Hope this helps.