Is the Product of Integrable Functions Also Integrable?

[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]

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 can't show that the integral exist because i don't 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]

can you, do,it if f,g are positive?

 

[quasar987]

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)