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 Incoming links: Strict inequality in Fatou lemma

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 27H / b / Solution Created 2026-09-24 Updated 2026-09-25

On the probability space

([0, 1], B, λ)

, let

f_{n} = n 1_{(0, 1/ n)} .

For every

x \in [0, 1]

, eventually

x \in / (0, 1/ n)

, so

f_{n} (x) \to 0

. Nevertheless,

\int_{0}^{1} f_{n} d λ = 1

for every

n

. Consequently

\int lim inf_{n} f_{n} d λ = 0 < 1 = lim inf_{n} \int f_{n} d λ .

This gives strict inequality in Fatou lemma.

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