I have the following problem:
Let $(\Omega,F,\mu)$ be a measure space. We define $x_A(w):=\left\{\begin{array}{ll} 1 & w \in A \\ 0 & w\notin A \\ \end{array}\right.$.
Is $f=y_1x_{A_1}+...+y_nx_{A_n}$ a staircase function with $y_i\geq0 \quad \forall i=1,...,n$ we define $$\int f d\mu:=y_1\mu(A_1)+...+y_n\mu(A_n)$$ as the integral of $f$.
Show that this integral is welldefined.
I guess I have to show it doesn't depend on the choices of the $A_i$. But I don't know how to do that exactly. Can someone help me?