Measure Theory/L^p spaces

Recall that an space is defined as Jensen's inequality Let be a probability measure space. Let , be such that there exist with If is a convex function on then, Proof Let . As is a probability measure, Let Let ; then Thus, , that is Put , which completes the proof. Corollary Putting , If is finite, is a counting measure, and if , then For every , define Holder's inequality Let such that . Let and . Then, and Proof We know that is a concave function Let , . Then That is, Let , , Then, , which proves the result Corollary If , then Proof Let , , Then, , and hence We say that if , almost everywhere on if . Observe that this is an equivalence relation on If is a measure space, define the space to be the set of all equivalence classes of functions in Theorem The space with the norm is a normed linear space, that is, for every , further, . . . (Minkowski's inequality) Proof 1. and 2. are clear, so we prove only 3. The cases and (see below) are obvious, so assume that and let be given. Hölder's inequality yields the following, where is chosen such that so that : Moreover, as is convex for , This shows that so that we may divide by it in the previous calculation to obtain . Define the space . Further, for define