Typical sets with an additional high-probability constraint ($|A^{n,\epsilon }\cap B_n|\ge 2^{n(H-\epsilon )-\delta }$)

Let $A^{n,\epsilon }$ be the weak typical set of a finite alphabet independent and identically distributed source with Shannon entropy $H$, and let $B_n$ be any sets with $\mathrm{Pr}(X^n\in B_n)\to 1$. The asymptotic equipartition property and union bound give $\mathrm{Pr}(X^n\in A^{n,\epsilon }\cap B_n)\to 1$. Every word in the intersection has probability mass between $2^{-n(H+\epsilon )}$ and $2^{-n(H-\epsilon )}$. Summing these bounds givesFor every fixed $\delta >0$, the first probability eventually exceeds $2^{-\delta }$, proving the title bound. A sample mean within a fixed positive tolerance of the expected value supplies $B_n$ by the strong law of large numbers.