Lebesgue Dominated Convergence Theorem

The dominated convergence theorem is a theorem which provides an additional condition under which pointwise convergence gives \(L_p\) convergence. This is typically done with \(L_1\) convergence as below, however beneath this result is a corollary which generalises it.

Theorem

Given a sequence of Lebesgue integrable functions \(\{f_{n}\}\) defined on a measure space \((X, \mathcal{F}, \mu)\) which satisfy:

  1. \(f_{n} \to f\) 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

\[ \lim_{n \to \infty} \|f_n - f\|_{L_1} = \lim_{n \to \infty} \int_{X} |f_{n} - f| \,\mathrm{d}\mu = 0.\]

Note that this final condition allows one to exchange the limit of the integral by applying the fact that \(L_1\) convergence implies integral convergence.

Corollary

Given \(\{f_n\}\) as above, with dominating function \(g \in L_p(X)\), we have that for any \(p \in [0, \infty)\)

\[ \lim_{n \to \infty} \|f_n - f\|_{L_p} = \lim_{n \to \infty} \left(\int_{X} |f_{n} - f|^p \,\mathrm{d}\mu\right)^{\frac{1}{p}} = 0.\]

Note this does not include \(L_{\infty}\), just finite \(p\).

Also note that previously we required our dominating function to be just integrable, that is, in \(L_1(X)\), to get convergence of the sequence in \(L_1\). Here, we similarly require the dominating function to be in \(L_p(X)\) for convergence of the sequence in \(L_p\).