Stochastic dynamics learning with state-space systems

The paper’s Theorem 3.8 proves stochastic ESP ⇒ stochastic FMP (and generic FMP on Polish spaces under a proper projection) by showing S_stoch is closed and invoking a general proposition on closed sets under proper projections; outputs follow by continuity of push-forwards. The candidate solution proves the same claims via a graph/projection argument for compact-valuedness and upper hemicontinuity, continuity of H_*, and a Baire-category argument for generic continuity. These are logically consistent and reach the same conclusions, though with slightly different routes. See the statement and proof sketch for Theorem 3.8 and the supporting proposition on upper hemi-continuity under properness in the paper , and the definitions of S_stoch/O_stoch and FMP/ESP , with continuity of push-forwards noted in Example 3.2 .