I have a question about this:
specifically I don't quite understand the meaning of $\mu(dx)$ in the integral in the question.
It says there is a measure space $(X, F, \mu)$ and a measurable function $f:X\rightarrow R$. Define the set $S$ as $S= \{x: f(x) >0\}$. Then it says suppose for some $B\in F$, this here holds $$\int_{S \cap B} f(x)\mu(dx)=0$$.
My question is: what is the meaning of the notation $\mu(dx)$. Because when I learned how to write integral of a function says $f(x)=x^2+3x+5$ and if I want to integrate it with respect to x from 0 to 2, I would do $$\int_{}(x^2+3x+5 )dx=0$$.
But I am not very sure what is the meaning and the purpose of $\mu(dx)$.
Is it because when I do the usual integral, there is no probability being assigned to the value in the domain?(i.e. there is no probability assigned to the x in my integral of $x^2+3x+5$). But when the integral was done in that measure space question, the $\mu(dx)$ indicates there is a probability assigned to each $x$, and the value of f at that particular x is being multiplied by the probability at that spot of x value?
$\int f(x)\mu(dx)$ is just another notation for $\int f(x) d\mu(x)$ or $\int f d\mu$, the integral of $f$ w.r.t. the measure $\mu$.