Existence of many finitely generated precompact subgroups of G and L_0(G)

For a Polish group G one can define L_0(G) as the set of all (Borel or) Lebesgue measurable functions from [0,1] to G. This set, after identifying functions that are equal up to a set of measure zero, when given the topology of convergence in measure and the group action of pointwise multiplication, becomes a Polish group. Although in some ways G and L_0(G) may differ pretty drastically - for example, L_0(G) is always (locally) path-connected, while  obviously G needs not to be - some of their properties are shared. We will show that among those is an existence of comeagerly many elements generating a precompact subgroup. In fact, we will show that if comeagerly n-tuples in G^n generate a precompact subgroup in G, then the same happens with respect to L_0(G).