convergence in measure

Convergence in measure asks a softer question than pointwise convergence. Instead of demanding that f_n(x) eventually settle near f(x) at each point, it only asks that the region where f_n is noticeably far from f keep shrinking — in size, as the measure sees it. A few points can misbehave at every stage, and even the misbehaving points can move around, as long as the total measure of the ‘bad’ region tends to zero.

Formally, on a measure space, f_n converges in measure to f if for every epsilon > 0 the measure of the set { x : |f_n(x) - f(x)| >= epsilon } tends to 0 as n tends to infinity. This is weaker than almost-everywhere convergence yet captures the essential idea of approximation that integration cares about. It is precisely the mode of convergence for which Markov-type estimates give control.

Its relations to other modes are worth memorizing. Convergence in L^p (for finite p) implies convergence in measure, by Chebyshev's inequality. Convergence in measure does not imply almost-everywhere convergence — but it always has a subsequence that does converge almost everywhere (the Riesz theorem). The classic illustration is the ‘typewriter sequence’ of indicator functions of shrinking, marching intervals on [0, 1]: it converges to 0 in measure, yet at no single point does it converge, because the moving bump revisits every point infinitely often.