Measure-theoretic question about a relationship between the pdf and the distribution function of a random variable For $(\Omega, \Bbb F, \Bbb P)$ a probability space and $X: \Omega \to \Bbb R$ a random variable, the distribution function of $X$ is defined as:
$$F_X(x) = \Bbb P(X^{-1}((-\infty,x]))$$
$X$ is called a continuous random variable if for some $f_X \in L^1(\Bbb R)$, such that $f_X \ge 0$, $F_X (x) = \int_{-\infty}^x f_X(t)dt$ for all $x$. In this case $f_X$ is called the pdf of $X$.
For a Borel set $B$, $\Bbb P(X \in B) := \Bbb P(X^{-1}(B))$.

If $X$ is a continuous random variable with pdf $f_X$, does the following hold for any Borel set $B$? Why?
$$\Bbb P(X \in B) = \int_B f_X(x)dx$$

This claim appears in a proof I am reading, and I don't understand why it holds. Thanks.
 A: Yes, that statement is true. One way to rectify this definition with your definition of the probability density of the random variable $X$ is to start with open intervals.  Suppose $B = (a,b)$.  Then, express $\Bbb{P}(X\in B)$ using the distribution function as 
$$
\Bbb{P}(X\in B) = \int_Bf_X(x)dx = \int_a^b f_X(t)dt = F_X(b) - F_X(a)
$$  Then, work up to a countable union of disjoint intervals.  Say $B = \cup_{j=1}^\infty(a_j,b_j)$.  By the countable additivity of measure, 
$$
\Bbb{P}(X\in \cup_{j=1}^\infty(a_j,b_j)) = \int_Bf_X(x)dx=\sum_{j=1}^\infty \int_{a_j}^{b_j}f_X(x)dx = \sum_{j=1}^\infty [F_X(b_j)-F_X(a_j)]
$$  Then, for any Borel set $B$, you have to use the (measure theory) fact that $B$ can be approximated by a countable union of disjoint intervals.  In fact, what you realize is that you are just constructing the definition of the Lebesgue integral of the function $\chi_B(x)f_X(x)\in L^1(\Bbb{R})$, where $\chi_B$ is the indicator function of $B$.  In fact, once you convince yourself that the two definitions are equivalent, you can just take 
$$
\Bbb{P}(X\in B) :=\int_{-\infty}^\infty \chi_B(x)f_X(x)dx
$$  as the definition of the probability measure for $X$.
A: There is a connection between the abs. continuity of a random variable ($X$) and the abs. continuity  of its distribution $P_X$ w.r.t. $m$ (the Lebesgue measure on $\mathbb{R}$). Namely, if $P_X$ is absolutely continuous w.r.t. $m$, then the Radon-Nikodym theorem implies that there exists $f_X=dP_X/dm$ (i.e. the R-N derivative of $P_X$ w.r.t. $m$), the pdf of $X$, such that for any Borel set $B$,
$$
\mathbb{P}\{X\in B\}=P_X(B)=\int_B f_Xdm,
$$
which is the result you're looking for.
