15 Jul '07 20:142 edits

okay, so this is the third and final question in my real (and abstract!) analysis exam this year. part i) is just a definition, part ii) requires a wise choice of partition and part iii) still evades me. and i got a good mark in the rest. it is just part iii). i wrote maybe a line, but i doubt i got any marks for it ðŸ˜›. so, can someone help me see part iii)? all of them need quite a good knowledge of the analysis branch of pure maths (specifically, integration).

so,

3. Let a, b be in the set R with a= R be a bounded function. Explain what it means for f to be integrable, and define the integral of f.

ii) Let f,q:[a,b] -> R be functions and let E be a finite subset of [a,b]. prove that f is integrable and q(x)=f(x) for all x in the set [a,b]\E, then q is integrable and int(q(t)dt,b,a) = int(f(t)dt,b,a).

iii) Let q:[a,b] -> R be a real function. Assume that for each n in the set N, the set (E subscript n)={x in the set[a,b] | |q(x)|>=(1/n)} is finite. Show that q is integrable and compute the integral int(q(t)dt,b,a).

[HINT: For each n in the set N consider the function (q subscript n):[a,b] -> R defined by (q subscript n)=

{0 if x is in the set [a,b]\(E subscript n)

{q(x) if x is in the set (E subscript n)

for x in the set [a,b]. Then use part (ii).]

anyone?

so,

3. Let a, b be in the set R with a= R be a bounded function. Explain what it means for f to be integrable, and define the integral of f.

ii) Let f,q:[a,b] -> R be functions and let E be a finite subset of [a,b]. prove that f is integrable and q(x)=f(x) for all x in the set [a,b]\E, then q is integrable and int(q(t)dt,b,a) = int(f(t)dt,b,a).

iii) Let q:[a,b] -> R be a real function. Assume that for each n in the set N, the set (E subscript n)={x in the set[a,b] | |q(x)|>=(1/n)} is finite. Show that q is integrable and compute the integral int(q(t)dt,b,a).

[HINT: For each n in the set N consider the function (q subscript n):[a,b] -> R defined by (q subscript n)=

{0 if x is in the set [a,b]\(E subscript n)

{q(x) if x is in the set (E subscript n)

for x in the set [a,b]. Then use part (ii).]

anyone?

*note:*int(f(t)dt,b,a) would be the integral of f(t) wrt t between b and a (with b at the top and a at the bottom).*noteII:*any reference to integration is referring to the riemann integral...