Fatou's lemma is a basic result on switching the order of limits and Lebesgue integrals, often proved as a consequence of the **monotone convergence theorem**. Its power lies in the fact that essentially *no* conditions are imposed on the sequence of functions; it is somewhat surprising that *anything* can be proved under these circumstances. On the other hand, under these conditions -- even the limit need not exist -- it is not surprising that equality cannot be proved.

**Lemma** (Fatou).

Let f_{1}(x), f_{2}(x), ... be any sequence of **measurable functions**. Then
∫ lim inf_{n→∞} f_{n}(x) dx ≤
lim inf_{n→∞} ∫ f_{n}(x) dx.

In particular, the integrand on the LHS is **measurable**, and if its integral is infinite then so is any limit of the integrals on the RHS.

Note that lim inf *always* exists, for any sequence of numbers. However, it is possible that it will be ∞; both sides of the inequality should be treated accordingly.

Consider now the case when, in fact, the limit

f(x) = lim_{n→∞} f_{n}(x)

exists

almost everywhere. Then

∫ f(x) dx ≤
lim inf_{n→∞} ∫ f_{n}(x) dx.

But if we look at -f

_{n} in the inequality above, we see also that

∫ f(x) dx ≥
lim sup_{n→∞} ∫ f_{n}(x) dx.

But, of course, for

*any* sequence lim inf a

_{n} ≤ lim sup, a

_{n}, with equality

iff the limit exists. Here we have the opposite ordering.
So in fact we see the basic property of Lebesgue integrals:

*If* a sequence of functions converges (

pointwise convergence,

almost everywhere), then the sequence of integrals converges to the integral of the sequences:

∫ f(x) dx **=**
lim_{n→∞} ∫ f_{n}(x) dx.

and the limit on the

RHS does, in fact, exist.