So I am a normal second year math student in who is currently taking probability theory. One of the things we learned is that if you have a continuous probability densities $f(x)$, then $\int_{-\infty}^{\infty}f(x)dx = 1$. We also learned that $P(X = x) = 0$ as well. So it means that the probability of something happening is 100%, but the probability of one specific thing is 0%. This is pretty mind blowing for me, but it brought me back to Calculus 1 where we defined integrals as the limit of a Riemann Sum (which made sense from a proof standpoint).
However now I seem to have trouble fundamentally grasping how an infinite sum of objects with zero area could have a value. I know that we are taking the limit, not the actual, but does that mean that if we ever reached 0 it would instantly disconnect and drop off to zero?
Thank you for any help!
You are summing "infinitely many" things with zero area, so that infinity and zero "compensate" each other.
Things become clearer when you think of a decomposition of $f$ in contiguous rectangles, thus approximating $f$ in a piecewise constant manner. When you shrink the rectangle width, their number increases, for an approximately constant sum.