94 1. Measure theory agree. Show that f is measurable. (This is a converse to Exercise 1.3.11.) We will continue to see the monotone convergence theorem, Fatou’s lemma, and the dominated convergence theorem appear throughout the rest of this text (and in An epsilon of room, Vol. I ). 1.5. Modes of convergence If one has a sequence x1,x2,x3,... ∈ R of real numbers xn, it is unambiguous what it means for that sequence to converge to a limit x ∈ R it means that for every ε 0, there exists an N such that |xn − x| ≤ ε for all n N. Similarly, for a sequence z1,z2,z3,... ∈ C of complex numbers zn converging to a limit z ∈ C. More generally, if one has a sequence v1,v2,v3,... of d-dimensional vec- tors vn in a real vector space Rd or complex vector space Cd, it is also unambiguous what it means for that sequence to converge to a limit v ∈ Rd or v ∈ Cd it means that for every ε 0, there exists an N such that vn−v≤ ε for all n ≥ N. Here, the norm v of a vector v = (v(1),...,v(d)) can be chosen to be the Euclidean norm v 2 := ( ∑d j=1 (v(j))2)1/2, the supre- mum norm v ∞ := sup1≤j≤d |v(j)|, or any other number of norms, but for the purposes of convergence, these norms are all equivalent a sequence of vectors converges in the Euclidean norm if and only if it converges in the supremum norm, and similarly for any other two norms on the finite- dimensional space Rd or Cd. If, however, one has a sequence f1,f2,f3,... of functions fn : X → R or fn : X → C on a common domain X, and a putative limit f : X → R or f : X → C, there can now be many different ways in which the sequence fn may or may not converge to the limit f. (One could also consider con- vergence of functions fn : Xn → C on different domains Xn, but we will not discuss this issue at all here.) This is in contrast to the situation with scalars xn or zn (which corresponds to the case when X is a single point) or vectors vn (which corresponds to the case when X is a finite set such as {1,...,d}). Once X becomes infinite, the functions fn acquire an infinite number of degrees of freedom, and this allows them to approach f in any number of inequivalent ways. What different types of convergence are there? As an undergraduate, one learns of the following two basic modes of convergence: (i) We say that fn converges to f pointwise if, for every x ∈ X, fn(x) converges to f(x). In other words, for every ε 0 and x ∈ X, there exists N (which depends on both ε and x) such that |fn(x)−f(x)| ≤ ε whenever n ≥ N.

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