1.5. Modes of convergence 97 We give four key examples that distinguish between these modes, in the case when X is the real line R with Lebesgue measure. The first three of these examples were already introduced in Section 1.4, but the fourth is new, and also important. Example 1.5.2 (Escape to horizontal infinity). Let fn := 1[n,n+1]. Then fn converges to zero pointwise (and thus, pointwise almost everywhere), but not uniformly in L∞ norm, almost uniformly in L1 norm, or in measure. Example 1.5.3 (Escape to width infinity). Let fn := 1 n 1[0,n]. Then fn con- verges to zero uniformly (and thus, pointwise, pointwise almost everywhere, in L∞ norm, almost uniformly, and in measure), but not in L1 norm. Example 1.5.4 (Escape to vertical infinity). Let fn := n1[ 1 n , 2 n ] . Then fn converges to zero pointwise (and thus, pointwise almost everywhere) and almost uniformly (and hence in measure), but not uniformly, in L∞ norm, or in L1 norm. Example 1.5.5 (Typewriter sequence). Let fn be defined by the formula fn := 1 [ n−2k 2k , n−2k+1 2k ] whenever k 0 and 2k n 2k+1. This is a sequence of indicator functions of intervals of decreasing length, marching across the unit interval [0, 1] over and over again. Then fn converges to zero in measure and in L1 norm, but not pointwise almost everywhere (and hence also not pointwise, not almost uniformly, nor in L∞ norm, nor uniformly). Remark 1.5.6. The L∞ norm f L∞(μ) of a measurable function f : X C is defined to the infimum of all the quantities M [0, +∞] that are essential upper bounds for f in the sense that |f(x)| M for almost every x. Then fn converges to f in L∞ norm if and only if fn f L∞(μ) 0 as n ∞. The L∞ and L1 norms are part of the larger family of Lp norms, studied in §1.3 of An epsilon of room, Vol. I. One particular advantage of L1 convergence is that, in the case when the fn are absolutely integrable, it implies convergence of the integrals, X fn X f dμ, as one sees from the triangle inequality. Unfortunately, none of the other modes of convergence automatically imply this convergence of the integral, as the above examples show. The purpose of these notes is to compare these modes of convergence with each other. Unfortunately, the relationship between these modes is not particularly simple unlike the situation with pointwise and uniform
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