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**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..