I was following the textbook by David Mackay: [*Information theory inference and learning algorithms*](http://www.cs.toronto.edu/~mackay/itprnn/book.pdf).

I have question on asymptotic equiparition' principle:


> For an ensemble of $N$ $i.i.d$ random variables $X^N=(X_1,X_2....X_N),$ with $N$ sufficiently large, the outcome $x=(x_1,x_2...x_N)$ is almost certain to belong to a subset of $|A_x^N|$ having only $2^{NH(x)}$ members, with each member having probability  "close-to" $2^{-NH(x)}$.


And then in the textbook, it also says that **typical set doesn't nesscarry to contain the most probable element set.** 

On the other hand, "smallest-sufficient set" $S_{\delta}$ which defines to be:

>the smallest subset of of $A_x$ satisfying  $P(x\epsilon S_{\delta})\ge 1-\delta $, for $0\leq{\delta}\leq1. $  In other words, $S_{\delta}$ is constructed by taking the most probable elements in $A_x$, then the second probable......until the total probabily is  $\ge1-{\delta}$.


My question is, **as $N$ increases, does $S_{\delta}$ approaches typical set such that these two sets will end up be equivalent of each other?** If the size of the typical set is identical to the size of $|S_{\delta}|$,  then why are we even bother with $S_{\delta}$? Why can't we just take the typical set as our compression scheme instead?