Multivariable Advanced Calculus

7.7. MEASURABLE FUNCTIONS 175Definition 7.7.2 Let (Ω, F, µ) be a measure space and let f : Ω → (−∞, ∞].Then f is said to be F measurable if any of the equivalent conditions of Lemma 7.7.1hold.Theorem 7.7.3 Let f n and f be functions mapping Ω to (−∞, ∞] where F is aσ algebra of measurable sets of Ω. Then if f n is measurable, and f(ω) = lim n→∞ f n (ω),it follows that f is also measurable. (Pointwise limits of measurable functions are measurable.)Proof: The idea is to show f −1 ((a, b)) ∈ F. Let V m ≡ ( a + 1 m , b − 1 m)and V m =[a +1m , b − 1 m]. Then for all m, Vm ⊆ (a, b) and(a, b) = ∪ ∞ m=1V m = ∪ ∞ m=1V m .Note that V m ≠ ∅ for all m large enough. Since f is the pointwise limit of f n ,f −1 (V m ) ⊆ {ω : f k (ω) ∈ V m for all k large enough} ⊆ f −1 (V m ).You should note that the expression in the middle is of the formTherefore,∪ ∞ n=1 ∩ ∞ k=n f −1k(V m).f −1 ((a, b)) = ∪ ∞ m=1f −1 (V m ) ⊆ ∪ ∞ m=1 ∪ ∞ n=1 ∩ ∞ k=nf −1k (V m)⊆ ∪ ∞ m=1f −1 (V m ) = f −1 ((a, b)).It follows f −1 ((a, b)) ∈ F because it equals the expression in the middle which is measurable.This shows f is measurable.Proposition 7.7.4 Let (Ω, F, µ) be a measure space and let f : Ω → (−∞, ∞].Then f is F measurable if and only if f −1 (U) ∈ F whenever U is an open set in R.Proof: If f −1 (U) ∈ F whenever U is an open set in R then it follows from the lastcondition of Lemma 7.7.1 that f is measurable. Next suppose f is measurable so thislast condition of Lemma 7.7.1 holds. Then by Theorem 5.3.10 if U is any open set inR, it is the countable union of open intervals, U = ∪ ∞ k=1 (a k, b k ) . Hencef −1 (U) = ∪ ∞ k=1f −1 ((a k , b k )) ∈ Fbecause F is a σ algebra.From this proposition, it follows one can generalize the definition of a measurablefunction to those which have values in any normed vector space as follows.Definition 7.7.5 Let (Ω, F, µ) be a measure space and let f : Ω → X where Xis a normed vector space. Then f is measurable means f −1 (U) ∈ F whenever U is anopen set in X.Now here is an important theorem which shows that you can do lots of things tomeasurable functions and still have a measurable function.Theorem 7.7.6 Let (Ω, F, µ) be a measure space and let X, Y be normed vectorspaces and g : X → Y continuous. Then if f : Ω → X is F measurable, it follows g ◦ fis also F measurable.