1. ## 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.

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

3. 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 $\displaystyle {f}_{n}$

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

5. 1-) i got it now when graphing. i used the measure of [0,1].
2-) it equals 0
thanks anyway.