Pointwise convergence implies convergence of the integrals
Is this true?
No — it is false.
Others that fail the same way
- Riemann integrable ⇒ continuous almost everywhere fails for… (i.e. 'bounded ⇒ integrable')
- A subset of ℝ with measure zero is countable
- |f| Riemann integrable ⇒ f Riemann integrable
- A composition of Riemann integrable functions is Riemann integrable
- fₙ → 0 pointwise on [0,1] ⇒ ∫₀¹ fₙ → 0
- If ∫₀^∞ f converges then f(x) → 0
From Lebesgue Measure and Integration › Lebesgue integral, MCT, DCT, Fatou