hmwk12sol_a

hmwk12sol_a - MATH 202A HOMEWORK 12 Exercise 4. Proof. Let...

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

View Full Document Right Arrow Icon
MATH 202A HOMEWORK 12 Exercise 4 . Proof. Let ˜ g ( x ) = R b 0 χ { x t } | f ( t ) | t dt . It is clear that the function in the integral is mea- surable and thus by Tonelli Theorem, we have R b 0 ˜ g ( x ) = R b 0 ± R b 0 χ { x t } | f ( t ) | t dt ² dx = R b 0 ± R b 0 χ { x t } | f ( t ) | t dx ² dt = R b 0 | f ( t ) | t ± R b 0 χ { x t } dx ² dt = R b 0 | f ( t ) | t ± R t 0 dx ² dt = R b 0 | f ( t ) | t t dt = R b 0 | f ( t ) | dt < . Hence, ˜ g is integrable and thus g ( x ) is also integrable. Now we can use Fubini Theorem and similarly, by replacing | f ( t ) | with f ( t ), we have R b 0 g ( x ) dx = R b 0 f ( t ) dt . ± Exercise 5 . (a) Proof. Since F is closed in R , x 0 ,y 0 , s.t. | x - x 0 | = δ ( x ) , | y - y 0 | = δ ( y ). Hence, | x - x 0 | ≤ | x - y 0 | , | y - y 0 | ≤ | y - x 0 | . By triangle inequality, we have | x - y 0 | = | x - y + y - y 0 | ≤ | x - y | + | y - y 0 | and thus | x - x 0 |-| y - y 0 | ≤ | x - y 0 |-| y - y 0 | ≤ | x - y | . Similarly, we have | y - y 0 |-| x - x 0 | ≤ | x - y | . Therefore, we know | δ ( x ) - δ ( y ) | = || x - x 0 | - | y - y 0 || ≤ | x - y | . ± (b) Proof. F c = t n N U n , where U n are disjoint open sets and the expression is unique. x F c , suppose x ( a,b ). Let I = ( x + a 2 , x + b 2 ). Hence, y I,δ ( y ) min { a - x 2 , b - x 2 } , r . Thus, we have I ( x ) R I δ ( y ) | x - y | 2 dy R I r | x - y | 2 dy r R ( a - x 2 , b - x 2 ) 1 z 2 dz . Since 0 ( a - x 2 , b - x 2 ), we have R ( a - x 2 , b - x 2 ) 1 z 2 dz = and thus I ( x ) ≥ ∞ , which implies I ( x ) = , x F c . ± (c) Proof. By hint, we investigate R F I ( x ) dx . Since δ ( x ) is continuous, δ ( y ) | x - y | 2 is also continuous and thus measurable. Hence, I ( x ) is non-negative and measurable function. By extended Fubini Theorem and part (a), we have R F I ( x ) dx = R F R F c δ ( y ) | x - y | 2 dydx = R
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.

Page1 / 4

hmwk12sol_a - MATH 202A HOMEWORK 12 Exercise 4. Proof. Let...

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