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I think the pages 249-250 are the most relevant source in ariane's pdf. Sierpinski outlines how to go from the cardinality hypothesis on $\vert \mathbb{R} \vert^{\aleph_0}$ to the existence of a non-measurable set, as per Ashutosh's précis, although without specific reference to the Lebesgue density theorem, which is formulated as an argument about symmetry. He notes that the proof does not require Zermeo's axiom (well-ordering principle). Sierpinski proves his result for any function $f(x)$ satisfying the properties $f(x) = f(y), x - y \in \mathbb{Q}, f(x) \neq f(y), x - y \notin \mathbb{Q}$.