  In this paper, we study the Onsager-Machlup function and its relationship to
the Freidlin-Wentzell function for measures equivalent to arbitrary infinite
dimensional Gaussian measures. The Onsager-Machlup function can serve as a
density on infinite dimensional spaces, where a uniform measure does not exist,
and has been seen as the Lagrangian for the ``most likely element". The
Freidlin-Wentzell rate function is the large deviations rate function for
small-noise limits and has also been identified as a Lagrangian for the ``most
likely element". This leads to a conundrum - what is the relationship between
these two functions?
  We show both pointwise and $\Gamma$-convergence (which is essentially the
convergence of minimizers) of the Onsager-Machlup function under the
small-noise limit to the Freidlin-Wentzell function - and give an expression
for both. That is, we show that the small-noise limit of the most likely
element is the most likely element in the small noise limit for infinite
dimensional measures that are equivalent to a Gaussian. Examples of measures
include the law of solutions to path-dependent stochastic differential
equations and the law of an infinite system of random algebraic equations.
