Every random variable satisfies a certain nontrivial integrability condition

We show that every random variable X fulfills a certain nontrivial integrability condition, in the sense that there always exists a nonnegative function g—depending on X  —growing to ∞∞ as x→∞x→∞, and such that Eg(|X|)<∞Eg(|X|)<∞. Refinements of this universal property allow us to give some simple but striking statements in connection with Markov's inequality and the central limit theorem.