Getting P(X=a) from a density curve

Question

If a density curve is described by $f(x)$, we know that we can get $P(x \ge a)$ or $P(x \le a)$ by calculating definite integrals of $f(x)$. Why cannot we obtain $P(x = a)$ by simply calculating $f(a)$? A standard reason that is provided is that it is nearly impossible to get $x=a$ samples. But what if we get such measurements?

Answer 1

When you are talking about a continuous random variable with a density function, it means that your probability measures is absolutely continuous with respect to Lebesgue measure. It means that your probability measure have at least the same sets of measure 0. The Lebesgue measure gives measure zero to singleton and your measure as well. (This is not completely formal, but I think it is sufficient) Thus, $f(a)\neq P(x=a)$.

Intuitively, you have so many possibilities that you cannot pick a particular value. Notice, that when you are talking about infinite countable random variable you can have a random variable that could give you infinity values but in this case you have that every singleton has a measure.

Discrete random variables are not absolutely continuous with respect to Lebesgue but they are absolutely continuous with respect to counting measure.