Basic question on countable intersection and union of sets I am just a beginner at measure theory, and I have a very basic question on the following fact found in Robert Ash, Probability and Measure theory , page 7:

Now this strikes me as a little asymmetric.
Very informally speaking, and I know this makes no sense, but we seem to have to have a concept of "$b^-$" in that say $[a,b)=a...b^-$, a concept of {b} in $[a,b]=[a,b) \cup \{b\}$, but no corresponding concept of "$a..b^+$" which I would have defined exactly as per the first equation $\cap_1^{+\infty} [a,b+ {1\over n}) $
I ask because in David Williams "Probability with Martingales" 3.12, "Skorokhod representation of a random variable with prescribed distribution function", the distributions are perforce right-continuous - really, this is enforced (in my mind as it stands) by the 'asymmetry' above, as exhibited by the first equation.
Put in another way which might make more sense:


*

*Is it possible to build a series of strictly decreasing sets $F_{n+1} \subset F_n$ such that the countable intersection $ \cap F_n=[a,b)$ ?

*(And just to make sure): Does anything change is we allow uncountable intersections in (1) above ?
Thank you in advance.

*

*Take $F_n=(a−1/n,b)$.

*I may be very ignorant of this, but I think uncountable intersection is not a thing. I have only encountered finite or infinite intersection, which compile the arbitrary intersection. I don't think you can define $F_n$ if the intersection is uncountable, since the notation implies that it is countable.
