Let be a –finite measure space and be measurable functions. Hölder’s inequality asserts for , one has
One standard proof is to integrate the AM-GM inequality
Here, I provide an alternative proof. First consider the case where is the characteristic function of a set of finite measure. Let
Remark: Note that if is bounded away from , we may improve the above argument. For instance, suppose for some and for all . Then
It follows that
Taking roots and letting establishes Hölder’s inequality in the case .
Remark: In many applications, (1) is enough. Nevertheless, extending to simple functions is an easy matter and Hölder’s inequality follows by the density of simple functions in . ♠