1.5. Modes of convergence 95 (ii) We say that fn converges to f uniformly if, for every ε 0, there exists N such that for every n N, |fn(x) f(x)| ε for every x X. The difference between uniform convergence and pointwise convergence is that with the former, the time N at which fn(x) must be permanently ε-close to f(x) is not permitted to depend on x, but must instead be chosen uniformly in x. Uniform convergence implies pointwise convergence, but not conversely. A typical example: the functions fn : R R defined by fn(x) := x/n converge pointwise to the zero function f(x) := 0, but not uniformly. However, pointwise and uniform convergence are only two of dozens of many other modes of convergence that are of importance in analysis. We will not attempt to exhaustively enumerate these modes here (but see §1.9 of An epsilon of room, Vol. I ). We will, however, discuss some of the modes of convergence that arise from measure theory, when the domain X is equipped with the structure of a measure space (X, B,μ), and the functions fn (and their limit f) are measurable with respect to this space. In this context, we have some additional modes of convergence: (i) We say that fn converges to f pointwise almost everywhere if, for (μ-)almost everywhere x X, fn(x) converges to f(x). (ii) We say that fn converges to f uniformly almost everywhere, essen- tially uniformly, or in L∞ norm if, for every ε 0, there exists N such that for every n N, |fn(x) f(x)| ε for μ-almost every x X. (iii) We say that fn converges to f almost uniformly if, for every ε 0, there exists an exceptional set E B of measure μ(E) ε such that fn converges uniformly to f on the complement of E. (iv) We say that fn converges to f in L1 norm if the quantity fn f L1(μ) = X |fn(x) f(x)| converges to 0 as n ∞. (v) We say that fn converges to f in measure if, for every ε 0, the measures μ({x X : |fn(x) f(x)| ε}) converge to zero as n ∞. Observe that each of these five modes of convergence is unaffected if one modifies fn or f on a set of measure zero. In contrast, the pointwise and uniform modes of convergence can be affected if one modifies fn or f even on a single point. The L1 and L∞ modes of convergence are special cases of the Lp mode of convergence, which is discussed in §1.3 of An epsilon of room, Vol. I. Remark 1.5.1. In the context of probability theory (see Section 2.3), in which fn and f are interpreted as random variables, convergence in L1 norm
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