Lebesgue Dominated Convergence Theorem

The dominated convergence theorem is a theorem which provides an additional condition under which pointwise convergence gives Lp convergence. This is typically done with L1 convergence as below, however beneath this result is a corollary which generalises it.

Theorem

Given a sequence of Lebesgue integrable functions {fn} defined on a measure space (X,F,μ) which satisfy:

  1. fn→f almost everywhere on X

  2. there exists an integrable function g such that


    |f_{n}| \leq g \quad \text{almost everywhere on} \ X.

Then, f is integrable and

limn→∞∥fn−f∥L1=limn→∞∫X|fn−f|dμ=0.

Note that this final condition allows one to exchange the limit of the integral by applying the fact that L1 convergence implies integral convergence.

Corollary

Given {fn} as above, with dominating function g∈Lp(X), we have that for any p∈[0,∞)

limn→∞∥fn−f∥Lp=limn→∞(∫X|fn−f|pdμ)1p=0.

Note this does not include L∞, just finite p.

Also note that previously we required our dominating function to be just integrable, that is, in L1(X), to get convergence of the sequence in L1. Here, we similarly require the dominating function to be in Lp(X) for convergence of the sequence in Lp.