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A question on the notation of an integral I discovered in my script:

$\int_G f(x) d\mu(x)$ is what I know. $\int_G d\mu(x) f(x)$ is what I read.

In the script $f(x)$ happens to be a characteristic function and $\mu$ the lebesgue measure, and we are using the lebesgue integral. Is this relevant?

Do these two notations mean the same thing? When you have multiple integrals within one another, using the second notation one would have to use brackets carefully to make clear which functions belongs to which integral I believe. What is the advantage or idea behind the second notation? Intuitively perhaps fubini has something to say?

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    They mean the same - putting $d \mu(x)$ first sometimes makes it clearer as to which variable(s) you are integrating against. This is especially useful if you're working in multiple variables – 2017-01-12
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    I don't understand, how is it clearer to put it before rather than directly after? You want to read first all the variables and then the function? – 2017-01-12

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In your case they are the same, but many times they are not, such as when $d\mu(x)$ has values on the set of linear operators acting some vector space $X$ and $f(x)$ takes value on $X$. Then the order is relevant.

This is common for example in the study of delay equations, when $d\mu$ corresponds to a Riemann-Stietjes integral (defined by some function with bounded variation) with values on $\mathbb R^n$ applying to $n\times n$ matrices.