# Lebesgue Integral

• May 22nd 2011, 06:01 AM
gilgames
Lebesgue Integral
hi, i need help for this question:
${f}_{n}(x)=n*{\varphi }_{[0,1/n]}$ where ${\varphi}_{[0,1/n]}$ is characterestic function.

1-)what is the value of $\int {f}_{n} d\mu$ over the interval [0,1] ?
2-) if n goes to infinity and ${f}_{n}(x)\to f(x)$ , what is the value of $\int f d\mu$ over the interval [0,1]

thanks for any help.
• May 22nd 2011, 06:03 AM
girdav
1) What is the integral of a characteristic function ?
2) Try to find $f$.
• May 22nd 2011, 06:12 AM
gilgames
1-)well the function ${f}_{n} (x)$ will be equal to $n$ for $x \epsilon [0,1/n]$ and otherwise it is 0.
so i find that $\int {f}_{n } d\mu = \int n d\mu$ over [0,1]. $\int nd\mu =n\int d\mu =n$ 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 ${f}_{n}$
• May 22nd 2011, 06:20 AM
girdav
1) No, we have $\int f_nd\mu =n\mu ([0,\frac 1n])=1$.
2) $f=0$ almost everywhere, so what about $\int fd\mu$ ?
• May 22nd 2011, 06:26 AM
gilgames
1-) i got it now when graphing. i used the measure of [0,1].
2-) it equals 0
thanks anyway.