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 f1(x), f2(x), ... be any sequence of measurable functions. Then
∫ lim infn→∞ fn(x) dx ≤
lim infn→∞ ∫ fn(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) = limn→∞ fn(x)
exists
almost everywhere. Then
∫ f(x) dx ≤
lim infn→∞ ∫ fn(x) dx.
But if we look at -f
n in the inequality above, we see also that
∫ f(x) dx ≥
lim supn→∞ ∫ fn(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 =
limn→∞ ∫ fn(x) dx.
and the limit on the
RHS does, in fact, exist.