density

let $\text{[math]}$ be a probability space, with $\text{[math]}$ . if there exists a nonnegative integrable function $\text{[math]}$ such that for any event $\text{[math]}$
$\text{[math]}$ then $\text{[math]}$ is called the probability density of $\text{[math]}$ .

it follows that the probability density must satisfy
$\text{[math]}$

[cite:@calc_hubbard_2015 definition 4.2.3]

if the distribution function $\text{[math]}$ is of the form
$\text{[math]}$ for some integrable function $\text{[math]}$ , then $\text{[math]}$ is called the density of the distribution.
[cite:@klenke_prob_2020 definition 1.106]

if the probability of a real-valued variable $\text{[math]}$ falling in the interval $\text{[math]}$ is given by $\text{[math]}$ for $\text{[math]}$ , then $\text{[math]}$ is called the probability density over $\text{[math]}$ . the probability that $\text{[math]}$ will lie in an interval $\text{[math]}$ is then given by
$\text{[math]}$ because probabilities are nonnegative, and because the value of $\text{[math]}$ must lie somewhere on the real axis, the probability density $\text{[math]}$ must satisfy the two conditions
$\text{[math]}$ [cite:;taken from @ml_bishop_2006 chapter 1.2.1 probability densities]