Monotone Convergence, Fatou and Dominated Convergence
Three theorems that say when a limit may be moved inside an integral, each with its own price. They are the reason this integral, and not the earlier one, is the one used in probability and in partial differential equations.
What a learner can do afterwards
- Apply monotone convergence and point at the step where monotonicity is used
- Give a sequence for which Fatou's inequality is strict
- Find a dominating function for a given sequence, or show that none exists
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When non-negative functions climb steadily, f1 below f2 below f3 and so on up to f, the integrals climb to the integral of f. That is monotone convergence. The climbing, the monotonicity, is the step the proof leans on: without it the conclusion can fail.
Fatou's lemma says that for non-negative f_n, the integral of the lower limit (the limit of tail infima) is at most the lower limit of the integrals. Let f_n be the indicator of [n, n+1], a bump of height 1 sliding right. Each integral is 1, but the pointwise limit is 0 with integral 0, so the inequality is strict. Tall spikes behave the same way: n on (0, 1/n) has integral 1 and pointwise limit 0.
A signed f is integrable when |f| has finite integral, and its integral is the integral of its positive part minus that of its negative part. Dominated convergence says the swap works when f_n converges to f and one integrable function g sits above every |f_n|. Then the integrals converge to the integral of f. The sliding bumps have no such g, since any cover of all bumps has infinite integral.
When a limit and an integral disagree, check whether the sequence climbs and whether one integrable function dominates it. The hypothesis that fails explains the gap.
Monotone convergence needs climbing, Fatou allows a gap, and dominated convergence needs one integrable ceiling, so check the price before swapping limit and integral.
Lesson, questions and flashcards are original text written for this dataset.
Reading notes and questions are original text written for this dataset, and flashcards were drafted from collected fact candidates.
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