Absolute Continuity of an Univariate Normal Random Variable

Question

Consider a normally distributed random variable $X\sim N(\mu,\sigma^2)$ and a Lebesgue measure $\lambda$ on $\mathbb{R}$.

Here, the value of the distribution at $\mu$, $F_X(\mu)$, is strictly positive, but the Lebesgue measure at $\mu$ is zero (i.e. $\lambda(\mu)=0$).

Then, there is a seemingly confusing point: the distribution $F_X$ is not absolutely continuous with respect the Lebesgue measure since $\lambda(\mu)=0$ does not imply $F_X(\mu)=0$, but, obviously the normal distribution has a density.

Where is the point of my misunderstanding?

Answer 1

The distribution $\eta_X$ of a random variable $X$ is a measure in $\mathbb R$. The function $F_X$ is not a measure, it is the cumulative distribution function.

For an interval $[a, b]$ the distribution gives the probability $\eta_X([a, b]) = F_X(b) - F_X(a)$. Thus, both the Lebesgue measure and the distribution $\eta_X$ have measure zero for the set $\{\mu\}$.

Answer 2

The probability density at 0 is positive, but the probability $P(X=0)$ is exactly zero -- as it has to be if there's a finite density.