# dG(y) in expected value integral

I am wondering what exactly the notation dG(y) inside an integral means, what it's called and where I can read more about it:

$$E[B_1]=\int_0^{\infty}E[B_1 \vert Y_1 = y] dG(y)$$

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It's the Stieltjes integral, and basically allows for the possibility that $G(x)$ may not have a continuous derivative -- for example, if $G$ is a discrete distribution.
If $G$ does have a continuous derivative, then your expression is the same as the usual Riemann integral
$$\int_0^{\infty}E[B_1 \vert Y_1 = y] g(y) dy$$
where $g(y) = dG(y)/dy$.