Monotone convergence and Fatou’s lemma

Suppose measurable functions fₙ are nonnegative and increase pointwise to f. Monotone convergence states that their integrals increase to the integral of f, with infinity allowed on both sides. There is no requirement that the space have finite measure or that a fixed integrable function dominate the sequence. The order condition prevents previously accumulated nonnegative mass from being removed. The theorem turns the lower-simple-function definition into a practical rule for constructing integrals by increasingly detailed approximations.

A useful consequence is exchanging a nonnegative series with integration: apply the theorem to its partial sums. This works even when the series or resulting integral diverges. If signs are present, partial sums may decrease and cancel; the theorem no longer applies directly. One must establish absolute integrability, a suitable dominating function, or some other theorem with the right assumptions. Naming the reason for the exchange is part of the calculation, especially when the final expression contains an infinite sum.

For a nonnegative sequence fₙ, define hₙ as the pointwise infimum of fₖ over all k≥n. Dropping early competitors can only increase an infimum, so hₙ increases to the pointwise lower limit of the sequence. Each hₙ is measurable and lies below every remaining fₖ. Monotone convergence applied to hₙ and monotonicity of integration then prove Fatou’s lemma. This construction shows exactly how an apparently unordered sequence is compared with an increasing family of reliable lower bounds.

Fatou’s direction says the integral of the pointwise lower limit cannot exceed the lower limit of the integrals. It does not assert equality and does not guarantee convergence of the original integrals. Nonnegativity can be replaced by a common integrable lower bound after subtracting that bound, but arbitrary negative parts are dangerous. On an infinite-measure space, subtracting a constant may create a nonintegrable adjustment, so the phrase bounded below must not be confused with bounded below by an integrable function.

Consider spikes n times the indicator of (0,1/n) on [0,1]. They tend pointwise to zero, including at zero because the interval excludes it, while each integral remains one. Fatou gives zero at most one, which is true and strict. The functions are not increasing: their heights rise while supports shrink. Nonnegativity alone therefore cannot justify equality. This example also rules out any common integrable dominating function, because such a function would permit a stronger limit theorem and contradict the computed areas.

A different mechanism occurs on the whole real line with indicators of [n,n+1]. Each integral is one and each fixed point eventually sees zero, but now mass escapes toward infinity rather than concentrating. This distinction matters when replacing domination by other conditions. A tail-height condition rules out tall spikes but does not by itself stop bounded mass from drifting across an infinite-measure domain. A limit theorem must control the actual mechanism by which an integral could differ from its pointwise prediction.