3zJv98-16

3zJv98-16 - MAT 473 39 16. Measurable Functions Recall that...

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MAT 473 39 16. Measurable Functions Recall that the σ -algebra B of Borel sets in R is the smallest σ -algebra of subsets of R which contains the open sets. Defnition 16.1. Let M be a σ -algebra of subsets of a set X . A function f : X R is said to be M -measurable if f 1 ( B ) M for each B B . Functions from R n into R which are B -measurable are called Borel measurable ; functions which are L -measurable are called Lebesgue measurable . Every Borel measurable function f : R n R is Lebesgue measurable, because B L . Example 16.2. For A X ,the characteristic function χ A of A is de±ned by χ A ( x )= ° 1i f x A 0i f x/ A. Since ( χ A ) 1 ( B )isa lwayse ither , A , A c ,or X , it is clear that χ A is M -measurable if and only if A M . In particular, there exist non-Lebesgue-measurable functions, and non-Borel-measurable Lebesgue-measurable functions, on R n . Example 16.3. Every continuous function f : R n R is Borel measurable. This will follow from Proposition 16.9, since f 1 ( U )isopenin R n for all open sets U R by continuity, and the open sets generate the Borel σ -algebra in R by de±nition. Proposition 16.4. If f : X R is M -measurable and g : R R is Borel measurable, then g f : X R is M -measurable. Proof. For each B B , g 1 ( B ) B , whence ( g f ) 1 ( B )= f 1 ( g 1 ( B )) M . ° There is a rough parallel between the continuous functions, which are those for which f 1 (

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3zJv98-16 - MAT 473 39 16. Measurable Functions Recall that...

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