202405291517
Status: #idea
Tags: Probability Theory, Measure Theory
State: #nascient

Fatou's Lemma

Let $A_{n}$ be a family of events.

$P (lim inf A_{n}) \leq lim inf P (A_{n}) \leq lim sup P (A_{n}) \leq P (lim sup A_{n})$
If $lim A_{n}$ exists then $lim P (A_{n}) = P (A)$ . In other words, if $A_{n} \to A ⟹ P (A_{n}) \to P (A)$ . In other words, P is continuous.

Preliminary Lemma:

$P (lim A_{n}) = lim P (A_{n})$ if $A_{n}$ is decreasing, or $A_{n}$ is increasing.
I leave this as a proof to future me.

Part 1

If you can prove the Lemma, then Fatou's Lemma first part follows after simple manipulation of the terms inside the parentheses. Hint (in what other ways can you write $lim i n f A_{n}$ and $lim s u p A_{n}$ )

Part 2

It follows pretty nicely from part $1$ .
We start by noticing that $lim A_{n} = A$ exists exactly when $lim inf A_{n} = lim sup A_{n}$ .

So we obtain:

A = lim A_{n} = lim inf A_{n} = lim sup A_{n}

Plugging that into part $1$ we get:

P (A) = P (lim inf A_{n}) \leq lim inf P (A_{n}) \leq lim sup P (A_{n}) \leq P (lim sup A_{n}) = P (A)

Since we squeezed the top and the bottom between $P (A)$ the chain of inequalities has to go to $P (A)$ .

And so it follows that as $A_{n} \to A$ , $P (A_{n}) \to P (A)$ .

Fatou's Lemma

Preliminary Lemma:

Part 1

Part 2

Relevant Links