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