Product of integrable functions

steven187

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

Science Advisor
Homework Helper
Gold Member
Notice that

$$fg=\frac{(f+g)^2-(f-g)^2}{4}$$

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.

steven187

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

Science Advisor
Homework Helper
can you, do,it if f,g are positive?

quasar987

Science Advisor
Homework Helper
Gold Member
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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