Under the generalized continuum hypothesis, the paper asserts every space with a free and transitive transformation group of size k^+ admits an invariant extension of its k-additive measurable structure containing 2^{k^+} sets pairwise separated in a uniformity.
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A Nonseparable Invariant Extension of Lebesgue Measure -- A Generalized and Abstract Approach
Under the generalized continuum hypothesis, the paper asserts every space with a free and transitive transformation group of size k^+ admits an invariant extension of its k-additive measurable structure containing 2^{k^+} sets pairwise separated in a uniformity.