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Laylah Mora

Laylah Mora

Answered question

2022-05-29

Why is μ ( { x X : | f ( x ) | > max { | z | : z essential range of  f } } ) = 0 for f L ( X , μ ) (i.e. the essential supremum of f is finite)?
I cannot find a good explanation here to be honest, help.

Answer & Explanation

glorietka4b

glorietka4b

Beginner2022-05-30Added 14 answers

Set max { | z | : z essential range of  f } = M < .
Then for each integer n, we have   μ ( | f | 1 ( ( M + 1 / n , ) ) ) = 0 and so by continuity of the measure, μ ( | f | 1 ( ( M , ) ) ) = 0.
And since { x X : | f ( x ) | > max { | z | : z essential range of  f } } | f | 1 ( ( M , ) ) it has measure zero too.

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