1. ## finding sequence of functions! difficult problem about integration..

Let $\displaystyle f,g$ are ANY non-negative and measurable. $\displaystyle \mu$ is measure
Find two function $f_n,g_n$ such that HAVE ONLY FINITELY MANY VALUE and
for any natural number $n$,
$\displaystyle |\int^{\infty}_0 \mu(\{ x : f(x) > t \}) dt - \int^{\infty}_0 \mu(\{ x : f_n(x) > t \}) | \leq \frac {C}{n}$ and
$\displaystyle |\int^{\infty}_0 \mu(\{ x : g(x) > t \}) dt - \int^{\infty}_0 \mu(\{ x : g_n(x) > t \}) | \leq \frac {C}{n}$ and
$\displaystyle |\int^{\infty}_0 \mu(\{ x : f(x) + g(x) > t \}) dt - \int^{\infty}_0 \mu(\{ x : f_n(x) + g_n(x) > t \}) | \leq \frac {C}{n}$.
And C is independent of n.

*Important* above integral is (improper) riemann integral !!
(In fact, this problem is from ANALYSIS by Lieb and Loss.. so definition of integral is different from other books
it defines integral by $\displaystyle \int f d\mu : = \int^{\infty}_0 \mu({x: f(x) > t}) dt$ so i wrote the problem as above..)
P.S. sorry for poor english..

-I correct some mistakes -

2. Originally Posted by ramsey88
Let f,g are non-negative and measurable. $\displaystyle \mu$ is measure
Find two function $\displaystyle f_n,g_n$ such that HAVE ONLY FINITELY MANY VALUE and
for any natural number n,
$\displaystyle |\int^{\infty}_0 \mu(\{ x : f(x) > t \}) dt - \int^{\infty}_0 \mu(\{ x : f_n(x) > t \}) | \leq \frac {C}{n}$ and
$\displaystyle |\int^{\infty}_0 \mu(\{ x : g(x) > t \}) dt - \int^{\infty}_0 \mu(\{ x : g_n(x) > t \}) | \leq \frac {C}{n}$ and
$\displaystyle |\int^{\infty}_0 \mu(\{ x : f(x) + g(x) > t \}) dt - \int^{\infty}_0 \mu(\{ x : f_n(x) + g(x) > t \}) | \leq \frac {C}{n}$.
And C is independent of n.

*Important* above integral is (improper) riemann integral !!
(In fact, this problem is from ANALYSIS by Lieb and Loss.. so definition of integral is different from other books
it defines integral by $\displaystyle \int f d\mu : = \int^{\infty}_0 \mu({x: f(x) > t}) dt$ so i wrote the problem as above..)
P.S. sorry for poor english..
I'm not clear on what the problem requires. Is there any requirement that would prevent a trivial example like $\displaystyle f_n(1)= 1$ and undefined for other x, for all n, $\displaystyle g_n(0)= 1$ and undefined for other x, for all n?

In that case, all integrals are 0 so you can take C to be any positive number.

3. Originally Posted by HallsofIvy
I'm not clear on what the problem requires. Is there any requirement that would prevent a trivial example like $\displaystyle f_n(1)= 1$ and undefined for other x, for all n, $\displaystyle g_n(0)= 1$ and undefined for other x, for all n?

In that case, all integrals are 0 so you can take C to be any positive number.
$\displaystyle f_n, g_n$ should satisfy above condition for any non-negative,measurable function f,g.