Let \((X,\mu)\) be a measure space and \(f_n, f\) measurable functions on \(X\). The following theorems justify \(\lim_n\int f_n = \int\lim_n f_n\) or an inequality between them.
Hypotheses and conclusions
| Theorem | Hypotheses | Conclusion |
|---|---|---|
| Monotone convergence | \(0 \leq f_1\leq f_2\leq\cdots\) and \(f_n \to f\) a.e. | \(\int f_n \to \int f\) |
| Fatou’s lemma | \(f_n \geq 0\) | \(\int \liminf f_n \leq \liminf \int f_n\) |
| Dominated convergence | \(f_n \to f\) a.e. and \(\abs{f_n}\leq g\) with \(g \in L^1\) | \(\int f_n \to \int f\) and \(\norm{f_n - f}_1 \to 0\) |
| Bounded convergence | \(\mu(X)<\infty\), \(\abs{f_n} \leq M\), and \(f_n \to f\) a.e. | \(\int f_n \to \int f\) |
Bounded convergence is dominated convergence with \(g\equiv M\), which is integrable if and only if \(\mu(X)<\infty\) (for \(M>0\)).
Choosing a theorem
Monotone nonnegative sequences. For measurable \(0\leq f_1\leq f_2\leq\cdots\), the monotone convergence theorem gives \(\int f_n\to\int\lim_n f_n\) in \([0,\infty]\), and for measurable \(g_k\geq0\) it gives \(\int\sum_k g_k = \sum_k\int g_k\).
Dominated sequences. The dominated convergence theorem requires an integrable \(g\) with \(\abs{f_n}\leq g\) for all \(n\). Common choices are a bound independent of \(n\) obtained from an explicit estimate, \(\sum_k\abs{h_k}\) when \(f_n\) are partial sums of \(\sum_k h_k\) with \(\sum_k\int\abs{h_k}<\infty\), and \(f_1\) when \(0\leq f_{n+1}\leq f_n\) and \(f_1\in L^1\).
Uniformly bounded sequences on a finite measure space. The bounded convergence theorem applies.
Nonnegative sequences without domination. Fatou’s lemma gives the inequality \(\int\liminf_n f_n\leq\liminf_n\int f_n\), and the inequality is strict for each sequence \(n\chi_{[0,1/n]}\), \(\frac1n\chi_{[0,n]}\), and \(\chi_{[n,n+1]}\).
Failure of the interchange
(Sequences with $f_n\to0$ pointwise and $\int f_n = 1$)
\(f_n \coloneqq n\chi_{[0,1/n]}\) on \([0,1]\): the mass concentrates near \(0\), and \(\sup_n f_n\notin L^1\).
\(f_n \coloneqq \frac1n \chi_{[0,n]}\) on \(\RR\): \(\abs{f_n}\leq1\), so bounded convergence fails on a space of infinite measure.
\(f_n \coloneqq \chi_{[n, n+1]}\) on \(\RR\): the mass moves to infinity.
In each case no integrable \(g\) dominates \((f_n)\), and \(0 = \int\lim_n f_n<\lim_n\int f_n = 1\).
Other interchanges
Commuting limits treats interchanges of limits more generally.
\(\sum_k \int f_k = \int \sum_k f_k\): by monotone convergence for \(f_k\geq0\), and by dominated convergence when \(\sum_k\int\abs{f_k} < \infty\); these are Tonelli’s and Fubini’s theorems for counting measure.
\(\frac{d}{dt}\int f(x,t)\,d\mu(x) = \int \partial_t f(x,t)\,d\mu(x)\): by dominated convergence applied to difference quotients, when \(\abs{\partial_t f(x,t)}\leq g(x)\) with \(g\in L^1\) for all \(t\) in an interval.
\(\int\int f\,d\mu\,d\nu\) in either order: Fubini and Tonelli.