Measure-theoretic question about a relationship between the pdf and the distribution function of a random variable

Question

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.

Answer 1

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 $X$.

Answer 2

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.