We derive Onsager--Machlup functionals for countable product measures on weighted $\ell^p$ subspaces of the sequence space $\mathbb{R}^{\mathbb{N}}$. Each measure in the product is a shifted and scaled copy of a reference probability measure on $\mathbb{R}$ that admits a sufficiently regular Lebesgue density. We study the equicoercivity and $\Gamma$-convergence of sequences of Onsager--Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussian, Cauchy, and Besov measures with summability parameter $1 \leq p \leq 2$. Together with Part I of this paper, this provides a basis for analysis of the convergence of maximum a posteriori estimators in Bayesian inverse problems and most likely paths in transition path theory.

翻译：我们从序列空间 $\ mathb{R ⁇ mathb{N ⁇ $ $ 的加权 $ = = p$ 子空间的可计算产品计量的 Onsager- Machlup 函数。 产品中的每一项计量都是 $\ mathbb{R} $ 的转换和缩放参考概率计量的复制件, 允许足够经常的 Lebesgue 密度 。 我们研究了与本类中措施的相趋同序列相关的Onsager- machlup 函数的均匀和 $\ gamma$- convergence 。 我们利用这些结果来为 separable Banach 或 Hilbert 空间的概率计量设定相似的结果, 包括 Gausian, Cauchy, 和 Besov 度的概率计量, 和可比较参数 1\leq p\leq 2$。 。 与本文第一部分一起, 这为分析Bayesian 问题和过渡路径理论中最有可能路径的最大匹配的远测算提供了基础。