$L_p$ spaces and tail estimates

Question

I can prove the main identity in this question. Not sure how the "and deduce" bit works. I think $O(\lambda^{-q})$ is some kind of tail estimate.

Answer 1

You are asked to prove that if $X\in L^q$ then there exists some finite $C$ such that, for every $\lambda\gt0$, $P(|X|\geqslant\lambda)\leqslant C\lambda^{-q}$. And that if this latter condition holds then $X\in L^p$.

Edit: It appears now that, in contradiction to the actual text of the question, the OP's problem is not to understand the meaning of these implications but to prove the first one. To do so, note that $$ E(|X|^q)\geqslant\int_0^\lambda qx^{q-1}P(|X|\geqslant x)\mathrm dx\geqslant\int_0^\lambda qx^{q-1}P(|X|\geqslant \lambda)\mathrm dx=\lambda^q P[|X|\geqslant\lambda). $$