Consider our reference set $\Omega = \mathscr{R}$ and our sigma-field is $F = \mathscr{B}$, the Borel Sigma Field, and suppose our measure is Lebesgue-Stieltjes $\mu$. Define $X_n$ as:
$$X_n = \frac{1}{n} \mathscr{1}_{[0,n]} $$
How to calculate $\int X_n d_\mu$ ? It's claimed that $\int X_n d_\mu = 1$ but I don't know how to get it.
Here is what I think:
$X_n$ is a simple function, therefore, the integral is equal to :
$\int X_n d_\mu = \sum_{0}^{\infty} 1/n* (n)$ but this is for sure not equal to 1.
thanks for your help.
I don't know what is leading you to write $\textstyle\sum_0^\infty$, but that is the source of your confusion. Recall that the integral of a simple function $$\sum_{i=1}^m c_i\mathbf{\large 1}_{A_i}$$ is, by definition, $$\sum_{i=1}^m c_i\mu(A_i).$$ In the simple function $X_n$, there is only one $c_i$ and $A_i$ (in other words, $m=1$). The value of $c_1$ is $\frac{1}{n}$ and the set $A_1$ is the interval $[0,n]$. Therefore $$\int X_n \,d\mu=\sum_{k=1}^1c_i\mu(A_i)=\frac{1}{n}\mu([0,n])=\frac{1}{n}\cdot n=1.$$