∑n∈N||fn−f||1<∞ implies fn converges almost uniformly to f, how to show this?
EDIT: Egorov's theorem is available. I have been able to show pointwise a.e. convergence using Chebyshev and Borel-Cantelli, I am having trouble trying to pass to almost uniform convergence using the absolute summability condition...
Answer
Put gn:=|fn−f|, and fix δ>0. We have ∑n∈N‖gn‖L1<∞ so we can find a strictly increasing sequence Nk of integers such that ∑n≥Nk‖gn‖1≤δ4−k. Put Ak:={x∈X:supn≥Nkgn(x)>21−k}. Then Ak⊂⋃n≥Nk{x∈X:gn(x)≥2−k} so
2−kμ(Ak)≤∑n≥Nk2−kμ{x∈X:gn(x)≥2−k}≤∑n≥Nk‖gn‖1≤δ4−k,
so μ(Ak)≤δ2−k. Put A:=⋃k≥1Ak. Then μ(A)≤∑k≥1μ(Ak)≤δ∑k≥12−k=δ, and if x∉A we have for all k: supn≥Nkgn(x)≤21−k so supn≥Nksupx∉Agn(x)≤21−k. It proves that gn→0 uniformly on Ac, since for a fixed ε>0, we take k such that 21−k, so for n≥Nk we have supx∉Agn(x)≤ε.
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