17 - 42 SPRING 2012 17. Measurable Functions, Simple...

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42 SPRING 2012 17. Measurable Functions, Simple Functions Proposition 17.1. If f,g : X R are M -measurable, then f + g and fg are M -measurable. Proof. For each t R , { x | f ( x )+ g ( x ) <t } = ° r Q ± { x | f ( x ) <r } { x | g ( x ) <t r } ² . Since f and g are M -measurable, each of the sets on the right-hand side is in M , so the left-hand set is as well. Thus f + g is M -measurable. Next, note that fg = ( f + g ) 2 ( f g ) 2 4 , so M -measurability of fg follows from the above and Proposition 16.6. ° Exercise 17.1. Prove that the σ -algebra B 2 of Borel sets in R 2 is generated by the collection S = { A × R | A B} { R × B | B B} , where B is the σ -algebra of Borel sets in R . (Note that the coordinate projections π i : R 2 R de±ned by π 1 ( x, y )= x and π 2 ( x, y )= y are continuous.) Conclude that if f,g : X R are M -measurable, then the function F : X R 2 de±ned by F ( x )=( f ( x ) ,g ( x )) is M -measurable. Use this to provide an alternative proof of Proposition 17.1. Proposition 17.2. If f n : X R is M -measurable for each n N , then so are sup n f n , inf n f n , lim sup n f n , lim inf n f n .
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17 - 42 SPRING 2012 17. Measurable Functions, Simple...

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