1.5. Modes of convergence 99 Fix ε 0, and let A := {x X : |f(x)−g(x)| ε}. This is a measurable set our task is to show that it has measure zero. Suppose for contradiction that μ(A) 0. We consider the sets AN := {x A : |fn(x) f(x)| ε/2 for all n N}. These are measurable sets that are increasing in N. As fn converges to f almost everywhere, we see that almost every x A belongs to at least one of the AN , thus N=1 AN is equal to A outside of a null set. In particular, μ( N=1 AN ) 0. Applying monotone convergence for sets, we conclude that μ(AN ) 0 for some finite N but by the triangle inequality, we have |fn(x)−g(x)| ε/2 for all x AN and all n N. As a consequence, fn cannot converge in measure to g, which gives the desired contradiction. 1.5.2. The case of a step function. One way to appreciate the distinc- tions between the above modes of convergence is to focus on the case when f = 0, and when each of the fn is a step function, by which we mean a con- stant multiple fn = An1En of a measurable set En. For simplicity we will assume that the An 0 are positive reals, and that the En have a positive measure μ(En) 0. We also assume the An exhibit one of two modes of behaviour: either the An converge to zero, or else they are bounded away from zero (i.e. there exists c 0 such that An c for every n). It is easy to see that if a sequence An does not converge to zero, then it has a subse- quence that is bounded away from zero, so it does not cause too much loss of generality to restrict to one of these two cases. Given such a sequence fn = An1En of step functions, we now ask, for each of the seven modes of convergence, what it means for this sequence to converge to zero along that mode. It turns out that the answer to the ques- tion is controlled more or less completely by the following three quantities: (i) The height An of the nth function fn (ii) The width μ(En) of the nth function fn and (iii) The N th tail support EN := n≥N En of the sequence f1,f2,f3,.... Indeed, we have: Exercise 1.5.3 (Convergence for step functions). Let the notation and assumptions be as above. Establish the following claims: (i) fn converges uniformly to zero if and only if An 0 as n ∞.
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