Fatou’s Lemma
Inequality for integrals of limits inferior.
Key idea
Inequality for integrals of limits inferior.
In depth
For non-negative fₙ, ∫ liminf fₙ ≤ liminf ∫fₙ. Mass can disappear in the limit but not appear from nowhere. It is the basic inequality behind other convergence theorems.
Worked example
fₙ = n·1_[0,1/n] → 0 pointwise (except 0) while ∫fₙ = 1, so ∫ liminf = 0 ≤ 1.
Real-world uses
- Bounding the expected value of limits in statistics and economic models
Common mistake
Expecting equality.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Fatou’s Lemma?
Inequality for integrals of limits inferior. It is a topic at the expert level in Analysis & Topology.
Where is Fatou’s Lemma used in real life?
Real-world examples: Bounding the expected value of limits in statistics and economic models.
What should I learn before Fatou’s Lemma?
Make sure you are comfortable with Dominated Convergence Theorem, the previous topic in Analysis & Topology.
What is a common mistake with Fatou’s Lemma?
Expecting equality.
How can I practise Fatou’s Lemma?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Fatou’s Lemma?
Move on to Fubini’s Theorem next.
Related expert topics in Analysis & Topology
- Axioms of the Real Numbers – ℝ is a complete ordered field.
- Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
- Archimedean Property – For any real x there is an integer n > x.
- Density of Rationals – Between any two reals there is a rational.
- Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
- Limit Theorems for Sequences – Algebra of limits and squeeze theorem.