- #1
scottneh
- 3
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Hello, I am preparing for a screening exam and I'm trying to figure out some old problems that I have been given.
Given:
Suppose [tex]\mu[/tex] is a finite Borel measure on R, and define
f[tex](x)=\int\frac{d\mu(y)}{\sqrt{\left|x-y\right|}}[/tex]
Prove f[tex](x)[/tex] is finite almost everywhere
If I integrate I get -2[tex]\sqrt{x-y}[/tex] for y[tex]\leq[/tex]x and 2[tex]\sqrt{-x+y}[/tex] for x[tex]\le[/tex]y
I can clearly see that the integral is not finite for x = y and hence the "almost everywhere constraint and the integrand looks like it would tend to zero as either variable increases toward infinity but the result of the integral looks like it will blow up if wither variable tend to infinity.
I was looking at Fubini-Tonelli but I'm not really sure how to use it here as only d mu(y) and not a d mu(x) as well.
Can someone please help me get started?
Thanks
Given:
Suppose [tex]\mu[/tex] is a finite Borel measure on R, and define
f[tex](x)=\int\frac{d\mu(y)}{\sqrt{\left|x-y\right|}}[/tex]
Prove f[tex](x)[/tex] is finite almost everywhere
If I integrate I get -2[tex]\sqrt{x-y}[/tex] for y[tex]\leq[/tex]x and 2[tex]\sqrt{-x+y}[/tex] for x[tex]\le[/tex]y
I can clearly see that the integral is not finite for x = y and hence the "almost everywhere constraint and the integrand looks like it would tend to zero as either variable increases toward infinity but the result of the integral looks like it will blow up if wither variable tend to infinity.
I was looking at Fubini-Tonelli but I'm not really sure how to use it here as only d mu(y) and not a d mu(x) as well.
Can someone please help me get started?
Thanks