Meaning of $\int_E {f(x) \mu(dx)}?$

Question

Suppose $f$ is a measurable real-valued function defined on a measure space $(E, X , \mu)$. What is the meaning of the RHS of the following integral

$$\int_E{f d\mu} = \int_E {f(x) \mu(dx)}?$$

I understand that LHS means 'integrate $f$ with respect to the measure $\mu$'. However, I fail to understand RHS.

Remark: The integral above is taken from here, under 'Construction - Integration'.

Answer 1

The notation $\mu(dx)$ seems to come from the Lebesgue integration $$ \sum f(\xi_i)\mu([x_i,x_{i+1}]) $$ where after taking the limit (plus translation invariance) the symbol $\mu(dx)$ appears.

The alternative notation $d\mu(x)$ looks more like to arrive from the Riemann-Stieltjes approach $$ \sum f(\xi_i)(g(x_{i+1})-g(x_i)) $$ where the limit gives $dg(x)$.

Another way to see the relation between those two notations (assuming everything to exist) $$ d\mu(x)=\mu'(x)dx=\tilde\mu(dx). $$

Answer 2

Suppose you're doing integration with the intuitive approach!

Then in Riemann Integral, you multiply the height of function in a very small part of horizontal line, called $dx$ !

But In Lebesgue Integral, The horizontal line must not be the real line, similarly the size of that small part ($dx$) is $\mu(dx)$ or in different notation $d\mu(x)$. Unfortunately, Lebesgue Integral doesn't have a standard notation as Riemann's Integral.

This is notation helps you, as well, to work with multiple integrals with different variables.

Answer 3

The RHS just emphasizes that $f$ and $\mu$ are functions on $X$. Sometimes $\int_E f d\mu$ is written $\int_E f(x) d\mu(x)$ as well.