Product of integrable functions

  • Thread starter steven187
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hello all

im in the middle of proving that if f and g are integrable functions then show that fg is also integrable

im up to trying to show that M_i(fg,P)-m_i(fg,P) is less than or equal to something that involves U(f,P)-L(f,P)<e^0.5 and U(g,P)-L(g,P)<e^0.5
anybody have any ideas, if i make any improvements I will post it up

thanxs
 

quasar987

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Notice that

[tex]fg=\frac{(f+g)^2-(f-g)^2}{4}[/tex]

So

[tex]\int_a^b fg dx = \int_a^b \frac{(f+g)^2-(f-g)^2}{4} dx[/itex]

if that second integral exists. Show that it does.
 
176
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hello there

well I have spent some time on it but, i cant show that the integral exist because i dont actually know what these functions are, I tried using it with the upper and lower sums but i aint getting anywhere that way

please help

thank you
 

mathwonk

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can you, do,it if f,g are positive?
 

quasar987

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Have you seen the theorem that say that if f and g and integrable, then af+bg (where a,b are constants) is integrable?

With that and the theorem that (basically) says that if F is integrable and G is continuous, then the composition G(F(x)) is integrable, you show that (f+g)² and (f-g)² are integrable (because x² is continuous and (f+g)² is the composition of f+g by x²)
 

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