Dominated convergence theorem

Henri Lebesgue's dominated convergence theorem states that if { f_n : n = 1, 2, 3, ... } is a sequence of measurable functions that converges almost everywhere, and the sequence is "dominated" (explained below) by some measurable function g whose integral is finite, then

$\int \lim _{n\rightarrow \infty }f_{n}=\lim _{n\rightarrow \infty }\int f_{n}.$

To say that the sequence is "dominated" by g means that

$-g(x)\leq f_{n}(x)\leq g(x)$

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.,

$\int \left|g\right|<\infty .$

Given the inequalities above, the absolute value sign enclosing g may be dispensed with.

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,

$\int \lim _{n\rightarrow \infty }f_{n}=0\neq 1=\lim _{n\rightarrow \infty }\int f_{n}.$