Likelihood-ratio vector for a point-mass model

Since $Q(\{j\})=2^{-j}$ and $P_k(\{j\})=1$ exactly when $j=k$, the density is $f_k(j)=2^k{1}_{\{j=k\}}$. Therefore $T(j)$ has coordinate $j$ equal to $2^j$ and all other coordinates zero.

The discrete metric on ${\mathbb{N}}_{\ge 1}$ is complete and separable, and its Borel sigma-field is the full sigma-field. The weights $2^{-k}>0$ sum to one, and step 1.1 gives finite nonnegative measurable density versions. Since every $Q$-atom has positive mass, the common $Q$-null set on which these versions are reset is empty. Thus all hypotheses of [F1], including AC [A1], hold, and that theorem proves minimal sufficiency. The vectors in step 1.1 are distinct for distinct $j$, proving injectivity. Finally, a set null under every $P_j$ contains no $j$, so the family-null ideal is trivial and the minimality comparison in [F2] is exact in this model.