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It is shown that the class of all non-separable extensions of a nonzeroσ   $\sigma $   -finite Borel measure in the topological vector spaceℝ  ℕ    $\mathbb {R}^{\mathbb {N}}$   , which are invariant under some everywhere dense continual subgroup ofℝ  ℕ    $\mathbb {R}^{\mathbb {N}}$    and which possess the uniqueness property, has maximal cardinality2   2  c$2^{2^{c}}$   . Some related questions concerning the measurability properties of real-valued functions with respect to the class of non-separable measures are also discussed.

On the uniqueness property of non-separable extensions of invariant Borel measures and relative measurability of real-valued functions