In Royden 4th P.36 it states that if you want to prove that finite union of measurable sets is measurable then you only have to prove that the union of two measurable sets is measurable. My question is if two measurable sets are measurable, can we have the conclusion that infinite union is also measurable?
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No , it's not true because induction doesn't apply to the infinite case – Tortar Jun 04 '20 at 10:29

See [here](https://math.stackexchange.com/questions/98093/whydoesntinductionextendtoinfinityrefourierseries]) – Tortar Jun 04 '20 at 10:31

Thanks for your answer @Tortar – yles will Aug 23 '21 at 03:01