Definitions

# Dominated convergence theorem

In measure theory, a branch of mathematical analysis, Lebesgue's dominated convergence theorem provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and pointwise convergence for a sequence of functions. This theorem shows the superiority of the Lebesgue integral over the Riemann integral for many theoretical purposes.

## Statement of the theorem

Let f1, f2, f3, ... denote a sequence of real-valued measurable functions on a measure space $\left(S,Sigma,mu\right)$. Assume that the sequence converges pointwise and is dominated by some integrable function g. Then the pointwise limit is an integrable function and

int_Slim_{ntoinfty} f_n,dmu=lim_{ntoinfty}int_S f_n,dmu. To say that the sequence is "dominated" by g means that

|f_n(x)| le g(x) for all natural numbers n and all points x in S. By integrable we mean

The convergence of the sequence and domination by g can be relaxed to hold only $mu$-almost everywhere.

## Proof of the theorem

Lebesgue's dominated convergence theorem is a special case of the Fatou–Lebesgue theorem. Below is a direct proof, using Fatou's lemma as the essential tool.

If f denotes the pointwise limit of the sequence, then f is also measurable and dominated by g, hence integrable. Furthermore,


|f-f_n|le 2g for all n and

limsup_{ntoinfty}|f-f_n|=0. By the reverse Fatou lemma,

limsup_{ntoinfty}int_S|f-f_n|,dmu leint_Slimsup_{ntoinfty}|f-f_n|,dmu=0. Using linearity and monotonicity of the Lebesgue integral,

biggl|int_Sf,dmu-int_Sf_n,dmubiggr| =biggl|int_S(f-f_n),dmubiggr| leint_S|f-f_n|,dmu, and the theorem follows.

## Discussion of the assumptions

That the assumption that the sequence is dominated by some integrable g can not be dispensed with may be seen as follows: define fn(x) = n for x in the interval (0,1/n] and fn(x) = 0 otherwise. Any g which dominates the sequence must also dominate the pointwise supremum h = supn fn. Observe that


int_0^1 h(x),dx geint_{1/m}^1 h(x),dx

# sum_{n

1}^{m-1}int_{left(frac1{n+1},frac1nright]}n,dx

# sum_{n

by the divergence of the harmonic series. Hence, the monotonicity of the Lebesgue integral tells us that there exists no integrable function which dominates the sequence on [0,1]. A direct calculation shows that integration and pointwise limit do not commute for this sequence:


int_0^1lim_{ntoinfty} f_n(x),dx

# 0neq 1

lim_{ntoinfty}int_0^1 f_n(x),dx,

because the pointwise limit of the sequence is the zero function.

## Extensions

The theorem applies also to measurable functions with values in a Banach space, with the dominating function still being non-negative and integrable as above.