If we pick a real number from $[0, 1]$ (with uniform probability measure), this number had probability zero to be picked.
Can we canonically assign an infinitesimal $\epsilon$ to the event $A_x$ that $x$ is picked such that, for example, $P(A_x \cup A_y) = 2\epsilon$ for $x \neq y$ or $P(A_{[a, b]}) = \epsilon + P(A_{(a, b]})$ (where $A_{[a, b]}$ and $A_{(a, b]}$ are the events that a number from $[a, b]$/$(a,b]$ is picked)?
This would amount to inserting more numbers into $[0, 1]$ and thus making the range of $P$ (our probability measure) larger.
The reason I ask is that such a model would perhaps take into account that something with probability zero might actually happen.