Dominated convergence theorem
In
mathematics,
Henri Lebesgue's
dominated convergence theorem states that if a sequence {
fn :
n = 1, 2, 3, ... } of
real-valued measurable functions on a
measure space S converges
almost everywhere, and is "dominated" (explained below) by some measurable function
g whose integral is finite, then
To say that the sequence is "dominated" by
g means that
for every
n and "almost every"
x (i.e., the
measure of the set of exceptional values of
x is zero). The theorem assumes that
g is "integrable", i.e.,
Given the inequalities above, the
absolute value sign enclosing
g may be dispensed with. In
Lebesgue's theory of integration, one calls a function "integrable" precisely if the integral of its absolute value is finite, so this hypothesis is expressed by saying that the dominating function is integrable.
That the assumption that the integral of g is finite cannot be dispensed with may be seen as follows: let f(x) = n if 0 < x < 1/n and f(x) = 0 otherwise. In that case,