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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?

Tobi92sr
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  • How are you defining the Lebesgue integral? – Math1000 May 26 '18 at 20:30
  • Show that if you have two representations of $f$ that the value of the integral is the same. One way is to pick a canonical representation and show equality. $f$ has finite range, so you can define some sets $B_k$ in terms of inverse images. – copper.hat May 26 '18 at 20:36
  • Hint: A canonical representation would make the $A_i$ disjoint. – Dzoooks May 26 '18 at 22:52
  • But what if I don't have these canonical representations? I found a proof here: https://proofwiki.org/wiki/Integral_of_Positive_Simple_Function_Well-Defined But they use that the $A_i$ are disjoint as well. How can I proof that if they aren't? for example : $$f=x_{]0,1]}+\frac{3}{2}x_{]1,2]}+\frac{1}{2}x_{]2,3]}=x_{]0,2]}+\frac{1}{2}x_{]1,3]}$$ – Tobi92sr May 27 '18 at 16:03

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