Purpose of Semidefinite Integral

Question

I want to know the meaning of Semidefinite Integral.

I am used to read definite and indefinite integral but I want to know the meaning of such equation :

$\pi(e)\left(1-F\left[-\frac{a}{\pi(e)}\right]\right) \left(a+\frac{\int_{-\frac{a}{\pi(e)}}\mu f(\mu)d\mu }{\left(1-F\left[-\frac{a}{\pi(e)}\right]\right)}\right)$

Where :

$\pi(e):$ is the probability of finding the information $\mu$ depending of the effort $e$ of the agent. $\left(1-F\left[-\frac{a}{\pi(e)}\right]\right)$ is the condition for the agent to search this information depending on the parameter $a$ which can be viewed as a reward.

$\mu$ is a stochastic term distributed by $f(\mu)$, the expected value of $\mu$ is 0.

I do not understand the second term of the equation where there is only the lower bound: $\int_{-\frac{a}{\pi(e)}}\mu f(\mu)d\mu$

Answer 1

It is just an opaque way to denote that the non-definite integration limit will take the value of the variable itself. Namely,

$$\int_a g(x) dx \equiv \int_a^x g(s)ds$$

... where $s$ is just the dummy variable of integration. Since sometimes we use the notation $\int _D f(x) dx$ to denote fully definite integration "over the domain $D$ of $x$", I would avoid the left-hand side notation and I would stick with the more clear right-hand-side one.

I do hope that the authors where you find this notation mean what I wrote above, because this is what mathematicians mean by it.