X is r.v. on ($\Omega$,F,P) where $E[\left\lvert X \right\rvert^a] < \infty$. Prove $E[\left\lvert X \right\rvert^b] < \infty$ for all $0<b\le a$

Question

X is r.v. on ($\Omega$,F,P) where $E[\left\lvert X \right\rvert^a] < \infty$. Prove $E[\left\lvert X \right\rvert^b] < \infty$ for all $0<b\le a$

How would I go about proving this using something in the realm of Lebesgue Integration or Domination thm. If I am off base with this please let me know. Having trouble even visualizing how to prove, would appreciate any help or hints.

edit: added abs value. sorry for that oversight!

Answer 1

Observe that $0<b\leq a$ implies that $|X|^b\leq 1+|X|^a$ and consequently: $$|\mathbb EX^b|\leq\mathbb E|X^b|=\mathbb E|X|^b\leq \mathbb E(1+|X|^a)=1+\mathbb E|X|^a<\infty$$

Actually the existence of $\mathbb EX^a$ allows us to conclude that $\mathbb E|X|^a=\mathbb E|X^a|<\infty$.