Convergence in L1 ( $E ∣ X_{n} - X ∣ \to 0$ )

Convergence in L1 means that the expected value of the absolute error tends to zero. It implies convergence in probability and convergence of expected values.

Scheffé lemma ( $f_{n} \to f a.e., \int f_{n} \to \int f ⟹ ∥ f_{n} - f ∥_{1} \to 0 (f_{n}, f \geq 0)$ )

For nonnegative integrable functions converging almost everywhere to an integrable

f

, convergence of their integrals implies L1 convergence. Indeed

min (f_{n}, f) \to f

almost everywhere and is dominated by

f

, so the dominated convergence theorem gives convergence of its integral. Use

∣ f_{n} - f ∣ = f_{n} + f - 2 min (f_{n}, f)

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