3zJv98-16

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 (
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This document was uploaded on 03/11/2012.

Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online