1.5. Modes of convergence 107 Finally, we recall two results from the previous notes for unsigned func- tions. Exercise 1.5.16 (Monotone convergence theorem). Suppose that fn : X → [0, +∞) are measurable, monotone non-decreasing in n and are such that supn X fn dμ ∞. Show that fn converges in L1 norm to supn fn. (Note that supn fn can be infinite on a null set, but the definition of L1 convergence can be easily modified to accommodate this.) Exercise 1.5.17 (Defect version of Fatou’s lemma). Suppose that fn : X → [0, +∞) are measurable, are such that supn X fn dμ ∞, and converge pointwise almost everywhere to some measurable limit f : X → [0, +∞). Show that fn converges in L1 norm to f if and only if X fn dμ converges to X f dμ. Informally, we see that in the unsigned, bounded mass case, pointwise convergence implies L1 norm convergence if and only if there is no loss of mass. Exercise 1.5.18. Suppose that fn : X → C are a dominated sequence of measurable functions, and let f : X → C be another measurable function. Show that fn converges pointwise almost everywhere to f if and only if fn converges almost uniformly to f. Exercise 1.5.19. Let X be a probability space (see Section 2.3). Given any real-valued measurable function f : X → R, we define the cumulative dis- tribution function F : R → [0, 1] of f to be the function F (λ) := μ({x ∈ X : f(x) ≤ λ}). Given another sequence fn : X → R of real-valued measurable functions, we say that fn converges in distribution to f if the cumulative distribution function Fn(λ) of fn converges pointwise to the cumulative dis- tribution function F (λ) of f at all λ ∈ R for which F is continuous. (i) Show that if fn converges to f in any of the seven senses discussed above (uniformly, essentially uniformly, almost uniformly point- wise, pointwise almost everywhere, in L1, or in measure), then it converges in distribution to f. (ii) Give an example in which fn converges to f in distribution, but not in any of the above seven senses. (iii) Show that convergence in distribution is not linear, in the sense that if fn converges to f in distribution, and gn converges to g, then fn + gn need not converge to f + g. (iv) Show that a sequence fn can converge in distribution to two differ- ent limits f, g, which are not equal almost everywhere. Convergence in distribution (not to be confused with convergence in the sense of distributions, which is studied in §1.13 of An epsilon of room, Vol. I, is commonly used in probability but, as the above exercise demonstrates,

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