 L1 convergence implies uniform integrability

ID: l1-convergence-implies-uniform-integrability

Z, Y

E [∣ Z ∣ 1_{{∣ Z ∣ > K}}] \leq 2 E ∣ Z - Y ∣ + E [∣ Y ∣ 1_{{∣ Y ∣ > K /2}}] .

Apply this with

Z = X_{n}

and

Y = X

to prove that convergence in L1 implies uniform integrability; control finitely many early terms separately.

 New to topics? Read the docs here!