
Lebesgue Integral
hi, i need help for this question:
$\displaystyle {f}_{n}(x)=n*{\varphi }_{[0,1/n]} $ where $\displaystyle {\varphi}_{[0,1/n]}$ is characterestic function.
1)what is the value of $\displaystyle \int {f}_{n} d\mu $ over the interval [0,1] ?
2) if n goes to infinity and $\displaystyle {f}_{n}(x)\to f(x)$ , what is the value of $\displaystyle \int f d\mu $ over the interval [0,1]
thanks for any help.

1) What is the integral of a characteristic function ?
2) Try to find $\displaystyle f$.

1)well the function $\displaystyle {f}_{n} (x)$ will be equal to $\displaystyle n$ for $\displaystyle x \epsilon [0,1/n] $ and otherwise it is 0.
so i find that $\displaystyle \int {f}_{n } d\mu = \int n d\mu $ over [0,1]. $\displaystyle \int nd\mu =n\int d\mu =n $ over [0,1].
is that right?
2) i can't figure how to find f. i sense f(x) must be zero when i graph $\displaystyle {f}_{n}$

1) No, we have $\displaystyle \int f_nd\mu =n\mu ([0,\frac 1n])=1$.
2) $\displaystyle f=0$ almost everywhere, so what about $\displaystyle \int fd\mu$ ?

1) i got it now when graphing. i used the measure of [0,1].
2) it equals 0
thanks anyway.