Strange notations in mathematical probability

I sometimes see things like $E[X] = \int t dF_X(t)$ or $E[X] = \int t F_X(dt)$ instead of the usual $E[X] = \int t f_X(t) dt$ . Where do these strange notations come from? Suppose we have a random variable $X$ , considered as a function which maps events from a $\sigma$ -field to an interval on the real line. Let the random variable take values in the range $[a, b]$ , which we will partition into $n$ subintervals $M_i = [b_{i-1}, b_i)$ , with $M_1 \equiv [b_0, b_1)$ , $M_2 \equiv [b_1, b_2)$ , $\ldots$ , $M_n \equiv [b_{n-1}, b_n]$ , and with $b_0 = a$ , $b_n = b$ . The random variable $X$ maps each event $E_i$ in the $\sigma$ -field to a corresponding interval $M_i$ on the real line according to a probability distribution. Pick an arbitrary point $t_i$ in each subinterval $M_i$ . Form a simple random variable $X_s$ from $X$ by assigning to each $E_i$ the value $X_s = t_i$ . Then the expectation of $X_s$ is given by

$E[X_s] = \sum_{i=1}^{n} t_i P(X_s = t_i) = \sum_{i=1}^{n} t_i P(X \in M_i)$

But if we denote the probability distribution function of $X$ by $F_X$ , we have

$P(X \in M_i) = F_X(b_i) - F_X(b_{i-1})$

Therefore we can write

$E[X_s] = \sum_i t_i (F_X(b_i) - F_X(b_{i-1})) \approx \int t dF_X(t)$

Therefore the notation $E[X] = \int t dF_X(t)$ means: the weighted sum of the t values, where the weights are the probabilities given by the difference in the values of the probability distribution function at the endpoints of the interval corresponding to t. But the approximations improve as the subdivisions become finer, so we can suppose $E[X]$ to be given by

$E[X] = \int t dF_X(t) = \int t F_X(dt)$

So the notation $\int t F_X(dt)$ means: the weighted sum of the t values, where the weights are the probabilities given by the difference in the values of the probability distribution function at the endpoints of the differential interval $dt$ .