Let fₙ = χ_{[n, n+1]} (the indicator function of the interval [n, n+1]) on ℝ with Lebesgue measure. Each fₙ has integral 1. What is ∫(liminf fₙ)?
A1, because every fₙ integrates to 1 and the liminf should preserve this
B∞, because infinitely many fₙ each contribute integral 1
C0, because for any fixed x the value fₙ(x) = 0 eventually, so liminf fₙ = 0 everywhere
D1/2, because the liminf averages the long-run behavior of the sequence
For any fixed x ∈ ℝ, once n > x we have x ∉ [n, n+1], so fₙ(x) = 0. Thus liminf fₙ(x) = 0 for every x, and the liminf function is identically zero. Its integral is 0. This is the canonical example of Fatou's strict inequality: ∫(liminf fₙ) = 0 ≤ 1 = liminf ∫fₙ. The mass did not disappear from the functions — it escaped to +∞, swept rightward out of every bounded region and therefore lost from the limiting function.
Question 2 Multiple Choice
Fatou's Lemma states ∫(liminf fₙ) ≤ liminf ∫fₙ. Why can the inequality be strict rather than an equality?
ABecause liminf is a strictly weaker operation than lim for sequences
BBecause mass can escape to infinity in the limit — the integral of the limiting function can be less than the limiting value of the integrals when mass 'runs away' to remote regions
CBecause the Lebesgue integral is not countably additive for infinite sequences of functions
DBecause non-negative functions do not have well-defined liminfs pointwise
The canonical example (indicator functions sweeping rightward) shows what strict inequality means: the functions always have integral 1, but in the limit no mass is captured — it has escaped to +∞. The liminf function is identically 0. This 'loss of mass' is real: the sequence of integrals stays at 1, but the integral of the limit function drops to 0. Fatou guarantees you can't gain mass in the limit; it cannot prevent you from losing it. The Dominated Convergence Theorem restores equality by imposing a dominating function that prevents this escape.
Question 3 True / False
Fatou's Lemma holds for any sequence of measurable functions, as long as the functions are measurable.
TTrue
FFalse
Answer: False
Non-negativity is an essential hypothesis, not a technical convenience. For functions taking negative values, the conclusion can fail dramatically. Consider gₙ = −χ_{[n,n+1]}: each integral is −1, but liminf gₙ = 0 everywhere (same reasoning as the positive case), giving ∫(liminf gₙ) = 0 > −1 = liminf ∫gₙ. The inequality reverses — the wrong direction entirely. Non-negativity is what prevents mass from escaping to −∞, where the lower-bound reasoning of Fatou's proof breaks down.
Question 4 True / False
Fatou's Lemma is primarily useful as a tool for directly computing integrals of limit functions.
TTrue
FFalse
Answer: False
Fatou's Lemma is a proof tool, not a computational one. It produces an inequality — ∫(liminf fₙ) ≤ liminf ∫fₙ — not an equality, so it cannot directly compute a limit integral. In practice, it appears in existence proofs (showing a limit function has a finite integral given bounds on the sequence's integrals) and in establishing lower semicontinuity of integral functionals. For actual computation of ∫(lim fₙ), you need Monotone Convergence or Dominated Convergence, which impose stronger conditions in exchange for an equality.
Question 5 Short Answer
State Fatou's Lemma informally and explain the intuitive reason why the inequality only goes one direction — why mass can be lost but not gained in the limit.
Think about your answer, then reveal below.
Model answer: For a sequence of non-negative measurable functions, the integral of the eventual lower envelope is at most the eventual lower limit of the integrals: ∫(liminf fₙ) ≤ liminf ∫fₙ. The inequality goes one way because mass can escape to infinity — as in functions that sweep rightward, carrying their mass to regions that no fixed point ever reaches — but mass cannot appear from nowhere in the limit. The limit function can only 'see' mass that stays in bounded regions; mass that runs to infinity is genuinely lost. Conservation holds in one direction: you cannot create mass in the limit. But you can lose it.
Think of it as a measure-theoretic conservation principle: the eventual lower envelope is the most mass the limit function can carry, and that is bounded above by how much mass was present in the sequence. The asymmetry (can lose, cannot gain) is why the inequality is ≤ rather than ≥ or =.