Is the following statement true? Pointwise convergence implies convergence of the integrals

No. fₙ = n·1_{(0,1/n)} on [0,1] fₙ → 0 pointwise but ∫fₙ = 1 always. Domination or monotonicity is essential.

Counterexample

Is this true?

Pointwise convergence implies convergence of the integrals

No — it is false.

The counterexample

$f_{n} = n\cdot1_{(0,1/n)}$ on [0,1]

Why it fails

$f_{n} \to 0$ pointwise but $\int{}f_{n} = 1$ always. Domination or monotonicity is essential.

Keep practicing

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.