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Following a similar result of Uspenskij on the unitary group of a separable Hilbert space, we show that, with respect to the lower (or Roelcke) uniform structure, the Polish group 
$G\,=\,\text{Aut(}\mu \text{)}$ of automorphisms of an atomless standard Borel probability space 
$(X,\,\mu )$ is precompact. We identify the corresponding compactification as the space of Markov operators on 
${{L}_{2}}(\mu )$ and deduce that the algebra of right and left uniformly continuous functions, the algebra of weakly almost periodic functions, and the algebra of Hilbert functions on 
$G$, i.e., functions on $G$ arising from unitary representations, all coincide. Again following Uspenskij, we also conclude that $G$ is totally minimal.
