Probability measure and distribution What is the difference between these two terms? From what I saw the two terms are not exactly interchangeable. I think that probability distribution implies probability measure, but the converse is not true. If this last statement is true, can you make an example?

Suppose that $(\Omega, \mathcal{S}, P)$ is a probability space and that
$X\rightarrow \mathbb{R}$ is a random variable.  You can define a borel measure $P_X$ on $\mathbb{R}$ by 
$$P_X(E) = P(X^{-1}(E)), \qquad E\in\mathcal{B}(\mathbb{R}).$$
This is the distribution of $X$. The cumulative distribution function is defined by 
$$F_X(x) = P_X((-\infty, x]),\qquad x\in \mathbb{R}.$$
Sometimes, this is just called the distribution function.  Every right continuous nondecreasing function $F:\mathbb{R}\rightarrow [0,1]$ with 
$F(\infty) = 1$ and $F(-\infty) = 0$ is a probability distribution. (for proof see Laha and Rohatgi's book).  
