Measure Theory/Integration

Let be a -finite measure space. Suppose is a positive simple measurable function, with ; are disjoint. Define Let be measurable, and let . Define Now let be any measurable function. We say that is integrable if and are integrable and if . Then, we write The class of measurable functions on is denoted by For , we define to be the collection of all measurable functions such that A property is said to hold almost everywhere if the set of all points where the property does not hold has measure zero. Properties Let be a measure space and let be measurable on . Then If , then If , , then If and then If , , then , even if If , , then , even if Proof Monotone Convergence Theorem Suppose and are measurable for all such that for every almost everywhere on Then, Proof is an increasing sequence in , and hence, (say). We know that is measurable and that . That is, Hence, Let Define ; . Observe that and Suppose . If then implying that . If , then there exists such that and hence, . Thus, , therefore . As this is true if , we have that . Thus, . Fatou's Lemma Let be measurable functions. Then, Proof For define . Observe that are measurable and increasing for all . As , . By monotone convergence theorem, and as , we have the result. Dominated convergence theorem Let be a complex measure space. Let be a sequence of complex measurable functions that converge pointwise to ; , with Suppose for some then and as Proof We know that and hence , that is, Therefore, by Fatou's lemma, As , implying that Theorem Suppose is measurable, with such that . Then almost everywhere Let and let for every . Then, almost everywhere on Let and then there exists constant such that almost everywhere on Proof For each define . Observe that but Thus for all , by continuity, almost everywhere on Write , where are non-negative real measurable.Further as are both non-negative, each of them is zero. Thus, by applying part I, we have that vanish almost everywhere on . We can similarly show that vanish almost everywhere on .