Hölder's inequality

There are several proofs of Hölder's inequality; the main idea in the following is Young's inequality for products.

hence the left-hand side is always bounded above by the right-hand side.

where we interpret any product with a factor of ∞ as ∞ if all factors are positive, but the product is 0 if any factor is 0.

are Hölder conjugates in (1, ∞). Application of Hölder's inequality gives

Proof of the conditional Hölder inequality: Define the random variables

and note that they are measurable with respect to the sub-σ-algebra. Since

the right-hand side is infinite and the conditional Hölder inequality holds, too. Dividing by the right-hand side, it therefore remains to show that

This is done by verifying that the inequality holds after integration over an arbitrary