I have somehow confused myself with this fairly straightforward proof. We need to show that &#x0

Crystal Wheeler 2022-07-02 Answered
I have somehow confused myself with this fairly straightforward proof. We need to show that λ f is a measurable function on ( S , S ), i.e. that for any c R : { s S : f ( s ) c } S. If λ 0, then the claim follows immediately from the measurability of f, namely as c / λ R it follows that { s S : f ( s ) c / λ } = { s S : λ f ( s ) c }
But then, if λ = 0 , λ f = 0 and I am not sure how to convince myself that λ f is measurable.
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Answers (2)

Aryanna Caldwell
Answered 2022-07-03 Author has 11 answers
If λ = 0, then your function λ f = 0 is identically 0. The inverse image of any measurable set containing 0 is S and of any set not containing 0 is the empty set. Both of these are measurable.
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aggierabz2006zw
Answered 2022-07-04 Author has 5 answers
f : S R measurable function.
g : R R defined by g ( x ) = λ x , λ R , is continuous.
Hence, g f = λ f is measurable.
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