Convergence Theorems

When may a limit move inside an integral? Monotone convergence, Fatou, and domination.

The idea

The Lebesgue convergence theorems answer one question: when may a limit be moved inside an integral, so that $\lim_{n} \int f_{n} = \int \lim_{n} f_{n}?$ The equality is not automatic — the two sides can differ — and each theorem states hypotheses under which it, or a weaker inequality, holds. In all three the $f_{n}$ are measurable functions on a measure space and the convergence is pointwise.

Theorem (Monotone Convergence Theorem (Lebesgue)).

If $f_{1}, f_{2}, \dots$ are measurable with $0 \le f_{1} \le f_{2} \le \cdots$ and $f_{n} \to f$ pointwise, then $\int f_{n} \to \int f$.

Lemma (Fatou's Lemma).

If $f_{1}, f_{2}, \dots$ are measurable with every $f_{n} \ge 0$, then $\int \liminf_{n} f_{n} \;\le\; \liminf_{n} \int f_{n}.$

Theorem (Dominated Convergence Theorem).

If $f_{1}, f_{2}, \dots$ are measurable with $f_{n} \to f$ pointwise, and there is a single integrable $g$ with $|f_{n}| \le g$ for every $n$, then $\int f_{n} \to \int f$.

The three theorems trade hypotheses for strength in different ways. Monotone convergence requires the sequence to be non-negative and increasing at every point, and concludes a full equality. Fatou's Lemma requires only non-negativity — not even convergence, since a $\liminf$ exists whether or not the sequence settles — and concludes only a one-way inequality, which can be strict. Dominated convergence drops monotonicity but requires a single integrable $g$ bounding every $|f_{n}|$ at once, and with that recovers the equality.

The domination hypothesis asks for one $g$ valid for all $n$; a separate bound for each $f_{n}$ proves nothing. Likewise the monotonicity hypothesis must hold at every point of the space, not merely on average. Whenever an exchange of limit and integral fails, one of these hypotheses is the one that failed.

Ways to work on it

• Walkthrough. Test the three convergence theorems on a sequence whose mass escapes to infinity.
• Proof. See why — derive Fatou's Lemma from the Monotone Convergence Theorem.
• Practice. Decide which theorem applies, or evaluate a limit one of them licenses.
• Hardest. Evaluate the limit of a sequence of integrals, justifying the exchange.

Not sure where to start? Take the ten-question placement test.