Let fₙ = n · 1_{[0,1/n]} on [0,1] with Lebesgue measure (a spike of height n and width 1/n). Each ∫fₙ dμ = 1 and the pointwise lim inf of fₙ is 0 everywhere. What does Fatou's Lemma say about this sequence?
A∫lim inf fₙ dμ = lim inf ∫fₙ dμ, so 0 = 1 — a contradiction showing the lemma fails here
B∫lim inf fₙ dμ ≤ lim inf ∫fₙ dμ, so 0 ≤ 1 — the inequality holds with strict inequality
C∫lim inf fₙ dμ ≥ lim inf ∫fₙ dμ, so 0 ≥ 1 — showing the functions are not admissible
DNothing — Fatou's Lemma only applies to monotone sequences
Fatou's Lemma guarantees ∫lim inf fₙ dμ ≤ lim inf ∫fₙ dμ. Here, lim inf fₙ = 0 pointwise (the spike concentrates in a region of width 1/n → 0), so ∫0 dμ = 0. And lim inf ∫fₙ = lim inf 1 = 1. We get 0 ≤ 1 — the inequality holds with strict inequality. This is the canonical example showing the inequality can be *strict*: mass has 'escaped to infinity' as the spikes narrow. Fatou does not claim equality; it only bounds the integral of the lim inf. Option D is wrong — Fatou requires only non-negativity and measurability, not monotonicity.
Question 2 Multiple Choice
Fatou's Lemma applies more broadly than the Monotone Convergence Theorem. What is the key difference in assumptions between the two results?
AThe MCT requires Lebesgue integration; Fatou's Lemma works for Riemann integration as well
BFatou's Lemma requires only non-negative measurable functions; the MCT additionally requires the sequence to be non-decreasing
CThe MCT applies to any sequence; Fatou's Lemma requires the sequence to converge pointwise
DFatou's Lemma requires a dominating integrable function; the MCT does not
The MCT requires the sequence to be non-decreasing (fₙ ≤ fₙ₊₁ a.e.) as well as non-negative, and under these conditions delivers equality: ∫lim fₙ = lim ∫fₙ. Fatou's Lemma requires only non-negativity — no monotonicity, no boundedness, no pointwise convergence — and delivers only an inequality. This weaker conclusion from weaker assumptions is what makes Fatou applicable in far more general situations. The dominating function assumption belongs to the Dominated Convergence Theorem, not Fatou.
Question 3 True / False
There exist sequences of non-negative measurable functions for which ∫lim inf fₙ dμ < lim inf ∫fₙ dμ strictly — Fatou's Lemma cannot be strengthened to an equality in general.
TTrue
FFalse
Answer: True
The moving spike example (fₙ = n · 1_{[0,1/n]}) is the canonical counterexample: each ∫fₙ = 1, so lim inf ∫fₙ = 1, while the pointwise lim inf is 0 everywhere, giving ∫lim inf fₙ = 0. Strict inequality holds because mass 'escapes to infinity' — the functions concentrate their mass in vanishingly small regions where the limit function has no mass. This shows that without additional assumptions (like domination by an integrable function), the inequality in Fatou cannot be improved to equality.
Question 4 True / False
Fatou's Lemma requires the sequence fₙ to converge pointwise to a function f before the inequality ∫lim inf fₙ dμ ≤ lim inf ∫fₙ dμ can be applied.
TTrue
FFalse
Answer: False
Fatou's Lemma requires only that the functions fₙ are non-negative and measurable — no pointwise convergence is needed. The lim inf in the statement handles the non-convergent case: lim inf fₙ(x) is defined for every sequence of real numbers, even if the sequence does not converge. The function g(x) = lim inf fₙ(x) is always well-defined and measurable for measurable fₙ. This is what makes Fatou so broadly applicable: it handles oscillating, non-converging sequences that the MCT cannot. If fₙ happens to converge pointwise, then lim inf fₙ = lim fₙ = f, and Fatou reduces to ∫f ≤ lim inf ∫fₙ.
Question 5 Short Answer
Explain why Fatou's Lemma cannot be upgraded to an equality in general — what phenomenon causes the strict inequality, and what additional assumption restores equality?
Think about your answer, then reveal below.
Model answer: The strict inequality arises when 'mass escapes to infinity' — when the sequence concentrates positive integral mass in regions that shrink to zero measure, so the limiting function captures none of that mass. The moving spike fₙ = n · 1_{[0,1/n]} illustrates this: each function has integral 1, but the limit function is 0 everywhere, so the integral of the limit is 0 while the limit of the integrals is 1. The mass that each fₙ carries is in a region of width 1/n that collapses to a single point of measure zero. To restore equality — guaranteeing ∫lim fₙ = lim ∫fₙ — one needs to prevent this escape. The Dominated Convergence Theorem provides this by requiring |fₙ| ≤ g a.e. for some integrable dominating function g; the dominating function bounds where mass can concentrate and prevents it from escaping to zero-measure sets.
Fatou only controls the 'floor' — what mass must survive to the limit. It cannot control mass that escapes to sets where the limit is zero. The DCT solves this by imposing a ceiling (the dominating function) that keeps mass in regions where convergence can be tracked, allowing equality to be restored.