Data assimilation (DA) in hidden Markov models evolves the filtering distribution via a nonlinear, measure-valued recursion that propagates uncertainty through the dynamics and updates it with observations. Classical ensemble methods such as the ensemble Kalman filter remain efficient and robust, but their Gaussian update can be inaccurate in strongly nonlinear, non-Gaussian regimes. We introduce the measure neural mapping (MNM), a learnable operator acting directly on probability measures to approximate the filtering map, leading to an MNM-enhanced ensemble filter (MNMEF) formulated both at the mean-field level and as an interacting-particle algorithm. MNMEF is implemented with the set transformer, enabling a single parameterization to transfer across ensemble sizes and the permutation invariance. We provide a theoretical foundation for this transfer by establishing a continuum limit for attention on measures: under mild regularity or boundedness assumptions, attention applied to empirical measures converges in Wasserstein distance to its continuous-measure counterpart as sample size increases. This continuum limit explains the observed stability of MNMEF trained on one ensemble size and evaluated across varying ensemble sizes. Empirically, MNMEF yields improved RMSE over leading baselines on Lorenz '96 and Kuramoto–Sivashinsky benchmarks for the state estimation problem.