PDF goes unbounded. Is probability of event infinite?

Question

This is follow up from here: Curve above $x$ axis but area is negative?

I have a PDF which has unit area but it goes unbounded to infinity at $x=b$ (please refer to attached link). Does it mean the probability of events near $b$ is infinite?

Answer 1

No. Assume $$f_X(x)=\begin{cases}-\ln x&,\quad 0< x\le 1\\0&,\quad \text{otherwise}\end{cases}$$then the CDF would be $$F_X(x)=\begin{cases} 0&,\quad x\le 0\\ x-x\ln x&,\quad 0< x\le 1\\1&,\quad x>1\end{cases}$$which is bounded, but of astonishingly high ascent near $x=0$.

Answer 2

Using axiomatic approach, probability is defined as Lebesgue measure $\mu$ on the subsets/intervals of a closed interval of real numbers, $[0,1]$, sample space $\Omega$. This measure/probability is equal to the length of such subset/interval. The 'smallest' Lebesgue measure is on isolated points ($0$), because the distance from the point to itself is $0$. The 'largest' Lebesgue measure is on the whole sample space $\Omega$. Again, by definition, it can't exceed $1$. So the probability/Lebesgue measure is strictly between $0$ and $1$.