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

<p>$$\int_a g(x) dx \equiv \int_a^x g(s)ds$$</p>

<p>... 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. </p>

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

CC BY-SA 4.0
2018-08-30T13:11:25.067Z
2018-08-30T16:31:53.803Z
2018-08-30T16:31:53.803Z
Alecos Papadopoulos
<p>I want to know the meaning of Semidefinite Integral.</p>

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

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

<p>Where :</p>

<p>$\pi(e):$ 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.</p>

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

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

<mathematical-economics><stochastic-processes>
Purpose of Semidefinite Integral
Alexandre
# Purpose of Semidefinite Integral

(Asked by: username_0 on 2018-08-30)

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:
$\int\_{-\frac{a}{\pi(e)}}\mu f(\mu)d\mu$

**Tags:** mathematical-economics, stochastic-processes

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## 1 answers

### Answer by username_1 on 2018-08-30. Score: 3 | Accepted answer

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.


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