80 1. Measure theory (viii) Show that the sum or product of two measurable functions in [0, +∞] or C is still measurable. Remark 1.4.33. One can also view measurable functions in a more category theoretic fashion. Define measurable morphism or measurable map f from one measurable space (X, B) to another (Y, C) to be a function f : X → Y with the property that f −1(E) is B-measurable for every C-measurable set E. Then a measurable function f : X → [0, +∞] or f : X → C is the same thing as a measurable morphism from X to [0, +∞] or C, where the latter is equipped with the Borel σ-algebra. Also, one σ-algebra B on a space X is coarser than another B precisely when the identity map idX : X → X is a measurable morphism from (X, B ) to (X, B). The main advantage of adopting this viewpoint is that it is obvious that the composition of mea- surable morphisms is again a measurable morphism. This is important in those fields of mathematics, such as ergodic theory (discussed in [Ta2009]), in which one frequently wishes to compose measurable transformations (and in particular, to compose a transformation T : (X, B) → (X, B) with itself repeatedly) but it will not play a major role in this text. Measurable functions are particularly easy to describe on atomic spaces: Exercise 1.4.30. Let (X, B) be a measurable space that is atomic, thus B = A((Aα)α∈I) for some partition X = α∈I Aα of X into disjoint non- empty atoms. Show that a function f : X → [0, +∞] or f : X → C is measurable if and only if it is constant on each atom, or equivalently if one has a representation of the form f = α∈I cα1Aα for some constants cα in [0, +∞] or in C as appropriate. Furthermore, the cα are uniquely determined by f. Exercise 1.4.31 (Egorov’s theorem). Let (X, B,μ) be a finite measure space (so μ(X) ∞), and let fn : X → C be a sequence of measurable functions that converge pointwise almost everywhere to a limit f : X → C, and let ε 0. Show that there exists a measurable set E of measure at most ε such that fn converges uniformly to f outside of E. Give an example to show that the claim can fail when the measure μ is not finite. In Section 1.3 we defined first a simple integral, then an unsigned in- tegral, and then finally an absolutely convergent integral. We perform the same three stages here. We begin with the simple integral: Definition 1.4.34 (Integral of simple functions). An (unsigned) simple function f : X → [0, +∞] on a measurable space (X, B) is a measurable

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