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The Three Convergence Theorems

Monotone convergence, Fatou's lemma, and dominated convergence — the reason the Lebesgue integral exists. Each says, under its own hypothesis, that the limit of the integrals equals the integral of the limit.

Why interchanging limit and integral is hard

We want to know when lim integral f_n = integral lim f_n — when we may pull a limit inside an integral. With the Riemann integral this almost always needed uniform convergence, a heavy hypothesis. The famous warning is the escaping bump: on [0, infinity) let f_n = 1 on [n, n+1] and 0 elsewhere. Then f_n -> 0 at every single point, yet integral f_n = 1 for all n. So lim integral f_n = 1 but integral lim f_n = 0. Mass leaked off to infinity. Any theorem must have a hypothesis ruling this out.

Monotone convergence and Fatou

[[monotone-convergence-theorem|Monotone Convergence Theorem]] (MCT). If 0 <= f_1 <= f_2 <= … are measurable and f_n increases pointwise to f, then integral f_n increases to integral f. The increasing hypothesis is what stops the escaping bump: a rising tide cannot leak mass away. MCT is the engine that powers additivity of the non-negative integral and, through that, all of linearity.

[[fatou-lemma|Fatou's Lemma]]. For any non-negative measurable f_n (no monotonicity, no convergence assumed), integral(liminf f_n) <= liminf integral f_n. It is the one-directional safety net: in the escaping-bump example liminf f_n = 0 so the left side is 0, while liminf integral f_n = 1 — the inequality 0 <= 1 holds, with strict loss exactly where mass escaped. Fatou follows from MCT applied to g_n = inf_{k>=n} f_k, which increase to liminf f_n.

Escaping bump:  f_n = 1 on [n, n+1], 0 elsewhere on [0, infinity)

  pointwise:  for each fixed x, f_n(x) = 0 once n > x   =>  f_n -> 0
  integrals:  integral f_n = 1  for every n

  MCT?    NO  -- f_n is not increasing (the bump moves), so MCT does not apply.
  Fatou:  liminf f_n = 0
          integral(liminf f_n) = 0   <=   liminf integral f_n = 1.   OK (strict).
  Dominated?  would need g >= |f_n| all n with integral g < infinity;
              the smallest such g is 1 on [0, infinity), integral = infinity.
              NO dominating majorant exists -> DCT does not apply either.
The escaping bump fails MCT and DCT but obeys Fatou — exactly the inequality, not equality.

The dominated convergence theorem

[[dominated-convergence|Dominated Convergence Theorem]] (DCT). Suppose f_n -> f almost everywhere, and there is a single fixed integrable majorant g — meaning |f_n| <= g for all n with integral g < infinity. Then f is integrable and lim integral f_n = integral f. Even more, integral |f_n - f| -> 0. This is the theorem you reach for in practice: no monotonicity, no uniform convergence needed — just pointwise convergence held in place by one integrable cap g.

Proof of DCT from Fatou (sketch):
  Hypotheses: f_n -> f a.e.,  |f_n| <= g,  integral g < infinity.

  Step 1.  g + f_n >= 0.  Apply Fatou to (g + f_n):
            integral(g + f) <= liminf integral(g + f_n)
            => integral g + integral f <= integral g + liminf integral f_n
            => integral f <= liminf integral f_n.
  Step 2.  g - f_n >= 0.  Apply Fatou to (g - f_n):
            integral(g - f) <= liminf integral(g - f_n)
            => integral g - integral f <= integral g - limsup integral f_n
            => limsup integral f_n <= integral f.
  Combine:  limsup integral f_n <= integral f <= liminf integral f_n,
            forcing  lim integral f_n = integral f.   QED
Two applications of Fatou (to g+f_n and g-f_n) squeeze limsup and liminf together — DCT is born.