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Suppose X is a Sierpinski set (So X is uncountable and every null subset of X is countable). Let f be a bijection on X. Must/Does there exist a non null subset Y of X such that for every subset W of Y, if W has same outer measure as Y, then f[W] has same outer measure as f[Y]?

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  • $\begingroup$ Maybe I'm missing something but, ae you assuming that $X$ is a subset of reals? If not, are you requiring any other kind of properties in its measure? $\endgroup$
    – Iván Ongay Valverde
    Commented Apr 28, 2015 at 19:27

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