88 1. Measure theory However, once one shuts down these avenues of escape to infinity, it turns out that one can recover convergence of the integral. There are two major ways to accomplish this. One is to enforce monotonicity, which pre- vents each fn from abandoning the location where the mass of the preceding f1,...,fn−1 was concentrated and which thus shuts down the above three escape scenarios. More precisely, we have the monotone convergence theo- rem: Theorem 1.4.43 (Monotone convergence theorem). Let (X, B,μ) be a mea- sure space, and let 0 ≤ f1 ≤ f2 ≤ . . . be a monotone non-decreasing sequence of unsigned measurable functions on X. Then we have lim n→∞ X fn dμ = X lim n→∞ fn dμ. Note that in the special case when each fn is an indicator function fn = 1En , this theorem collapses to the upwards monotone convergence property (Exercise 1.4.23(ii)). Conversely, the upwards monotone conver- gence property will play a key role in the proof of this theorem. Proof. Write f := limn→∞ fn = supn fn, then f : X → [0, +∞] is measur- able. Since the fn are non-decreasing to f, we see from monotonicity that X fn dμ are non-decreasing and bounded above by X f dμ, which gives the bound lim n→∞ X fn dμ ≤ X f dμ. It remains to establish the reverse inequality X f dμ ≤ lim n→∞ X fn dμ. By definition, it suﬃces to show that X g dμ ≤ lim n→∞ X fn dμ, whenever g is a simple function that is bounded pointwise by f. By vertical truncation we may assume without loss of generality that g also is finite everywhere, then we can write g = k i=1 ci1Ai for some 0 ≤ ci ∞ and some disjoint B-measurable sets A1,...,Ak, thus X g dμ = k i=1 ciμ(Ai).

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