1.4. Abstract measure spaces 79 (i) If μ(X) ∞, show that for every E A and ε 0 there exists F A such that μ(EΔF ) ε. (ii) More generally, if X = n=1 An for some A1,A2,... A with μ(An) for all n, E A has finite measure, and ε 0, show that there exists F A such that μ(EΔF ) ε. 1.4.4. Measurable functions, and integration on a measure space. Now we are ready to define integration on measure spaces. We first need the notion of a measurable function, which is analogous to that of a continuous function in topology. Recall that a function f : X Y between two topo- logical spaces X, Y is continuous if the inverse image f −1(U) of any open set is open. In a similar spirit, we have Definition 1.4.32. Let (X, B) be a measurable space, and let f : X [0, +∞] or f : X C be an unsigned or complex-valued function. We say that f is measurable if f −1(U) is B-measurable for every open subset U of [0, +∞] or C. From Lemma 1.3.9, we see that this generalises the notion of a Lebesgue measurable function. Exercise 1.4.29. Let (X, B) be a measurable space. (i) Show that a function f : X [0, +∞] is measurable if and only if the level sets {x X : f(x) λ} are B-measurable. (ii) Show that an indicator function 1E of a set E X is measurable if and only if E itself is B-measurable. (iii) Show that a function f : X [0, +∞] or f : X C is measurable if and only if f −1(E) is B-measurable for every Borel-measurable subset E of [0, +∞] or C. (iv) Show that a function f : X C is measurable if and only if its real and imaginary parts are measurable. (v) Show that a function f : X R is measurable if and only if the magnitudes f+ := max(f, 0), f− := max(−f, 0) of its positive and negative parts are measurable. (vi) If fn : X [0, +∞] are a sequence of measurable functions that converge pointwise to a limit f : X [0, +∞], then show that f is also measurable. Obtain the same claim if [0, +∞] is replaced by C. (vii) If f : X [0, +∞] is measurable and φ: [0, +∞] [0, +∞] is continuous, show that φ f is measurable. Obtain the same claim if [0, +∞] is replaced by C.
Previous Page Next Page