Invariant metric under deformed Markov embeddings with overlapped supports

Due to Čencov's theorem, there exists a unique family of invariant symmetric $(0,2)$-tensor fields on the space of positive probability measures on a set of $n$-points indexed by $n\in \mathbb{N}$ under Markov embeddings. We deform Markov embeddings keeping sufficiency, and prove existence and uniqueness of invariant families under the embeddings.

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