I try to understand arXiv:1909.03562 by Terence Tao. He uses the following definition of "almost all":
For a finite not empty subset $R$ of $\mathbb{N}$, $\text{Log}(R)$ is defined to be a random variable taking values in $R$ with logarithmically uniform distribution, so we have \begin{align} \mathbb{P}(\text{Log}(R) \in A) =\frac{\sum_{N \in A \cap R}{\frac{1}{N}}}{\sum_{N \in R}{\frac{1}{N}}}\end{align}
With this, the logarithmically density of $A \subset \mathbb{N}$ is defined as $\lim_{x \to \infty} \mathbb{P}(\text{Log}(\mathbb{N} \cap[1, x]) \in A)$.
Then a property $P$ holds for almost all $N$ if $\lim_{x \to \infty}\mathbb{P}(P(\text{Log}(\mathbb{N} \cap[1, x])) )=1$, so if it holds on a set of logarithmically density $1$.
As I have only little experience with probability theory, I do not have an intuition for this definition. How does it relate to the "classical" definition of "almost all": Holding for all, but finitely many elements?
I think, that if a statement holds for almost all in the classical definition it also holds in this sense, but what about the other direction? How large can the size of the set of exceptions get and how does this all relate to logarithms?
Thank you in advance for your help.