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How do we show that F(x) is continuous? (f is Lebesgue integrable on [a,b] )

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- Thread starter math8
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- #1

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How do we show that F(x) is continuous? (f is Lebesgue integrable on [a,b] )

- #2

quasar987

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[tex]F(x+h)-F(x)=\int_a^b\chi_{(x,x+h)}f(t)dt[/tex]

for h>0, and

[tex]F(x+h)-F(x)=-\int_a^b\chi_{(x+h,x)}f(t)dt[/tex]

for h>0, and use Lebesgue Dominated Convergence theorem to show that F(x+h)-F(x)-->0.

- #3

Hurkyl

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[tex]F(x+h) - F(x) = \int_x^{x+h} f(t) \, dt[/tex]

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lim integral =integal lim

but does lim K(x+h,x)*f(t) tends to 0? If yes, why?

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lim integral =integal lim

but does lim K(x+h,x)*f(t) tends to 0? If yes, why?

As far as I understand the limit of the indicator function will tend to 0 which in return will make the integral 0.

- #6

quasar987

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[tex]\chi_{(x,x+h)}\rightarrow\chi_{\emptyset}[/tex]

([itex]\chi_{\emptyset}[/itex] is just the function that is identically 0).

In a way, Hurkyl's way is swifter if you know that in a finite measure space X (such as [a,b] with the Lebesgue measure), for any [itex]1\leq r\leq s\leq +\infty[/itex], [itex]L^s(X)\subset L^r(X)[/tex]. Because then you can just write

[tex]|F(x+h)-F(x)|\leq \int_x^{x+h}|f(t)|dt\leq||f||_{\infty}(x+h-x)\rightarrow 0[/tex]

- #7

Hurkyl

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- #8

quasar987

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Scratch what I said about Hurkyl's idea.

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