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.
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