What does a greater than sign mean when it is a subscript of an integral sign?

Question

In a research paper that I am reading, I see the following:

$\int_{u_j > |\beta_j|} e^{-(\lambda_1u_j + \lambda_2\beta_j^2)}du_j$

What does the subscript of the integral sign mean? Is this a condition? Thanks!

Answer 1

You can replace it with $$\int_{\mid\beta_j\mid}^{\infty}\ldots du_j$$

Answer 2

Generally, if $x \in U$, then an integral can be written as $\int_\Omega f(x) dx$, where $\Omega \subset U$. Sometimes, as an abuse of notation, some statement $S(x)$ will be used in place of $\Omega$, i.e. $\int_{S(x)} f(x) dx$. In this case, simply define $\Omega = \{ x\in U \ | \ S(x) \}$, and $\int_{S(x)} f(x) dx := \int_{\Omega} f(x) dx$.