Fatou’s Lemma – Visual & Verbal Module
Heuristic Intuition
Imagine tracking a moving line — each approximation riding just above a ground curve.
Fatou’s Lemma says: the area under the ultimate limit curve can never exceed the lowest boundary traced by the sequence of previous areas.
Plain Statement
For non-negative measured quantities:
“The total of what you eventually get is no more than the least of what you could have gotten.”
“The total of what you eventually get is no more than the least of what you could have gotten.”
Why It Matters
This gives us a safety net for limit processes — it guarantees that integrals (or expected values) won’t suddenly spike when passing to the limit.
Use It When…
✅ You have a sequence of approximating functions
✅ You want to switch the limit with an integral
✅ You can’t use Dominated Convergence yet
✅ You want to switch the limit with an integral
✅ You can’t use Dominated Convergence yet
Zorn’s Lemma
Zorn’s Lemma:
If every chain (totally ordered subset) in a partially ordered set has an upper bound, then the set contains at least one maximal element.
If every chain (totally ordered subset) in a partially ordered set has an upper bound, then the set contains at least one maximal element.
A chain is a subset in which any two elements are comparable.
The animated rising path in the diagram shows a chain climbing through the poset.
If every chain has an upper bound, the poset can’t “escape upward forever.”
This ensures the existence of stopping points that become candidates for maximal elements.
A maximal element is one that has no strictly larger element above it.
In the animation, the glowing top node represents such a maximal point.
Zorn’s Lemma, the Well-Ordering Theorem, and the Axiom of Choice are all
logically equivalent.
Accepting one grants the others.
You must be logged in to post a comment.