106 1. Measure theory and |fn−f|κ |f|≤δ/2 |f| dμ ≤ ε and hence by the triangle inequality (1.17) |fn−f|κ |f|≤δ/2 |f − fn| dμ ≤ 2ε. Finally, from Markov’s inequality (Exercise 1.4.35(vi)) we have μ({x : |f(x)| δ/2}) ≤ A δ/2 and thus |fn−f|κ |f|δ/2 |f − fn| dμ ≤ ε ≤ A δ/2 κ. In particular, by shrinking κ further if necessary we see that |fn−f|κ |f|δ/2 |f − fn| dμ ≤ ε and hence by (1.17) (1.18) |fn−f|κ |f − fn| dμ ≤ 3ε for all n. Meanwhile, since fn converges in measure to f, we know that there exists an N (depending on κ) such that μ(|fn(x) − f(x)| ≥ κ) ≤ κ for all n ≥ N. Applying Exercise 1.5.13, we conclude (making κ smaller if necessary) that |fn−f|≥κ |fn| dμ ≤ ε and |fn−f|≥κ |f| dμ ≤ ε and hence by the triangle inequality |fn−f|≥κ |f − fn| dμ ≤ 2ε for all n ≥ N. Combining this with (1.18) we conclude that fn − f L1(μ) = X |f − fn| dμ ≤ 5ε for all n ≥ N, and so fn converges to f in L1 norm as desired.

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright no copyright American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.