I am studying Richard Laver's "On the consistency of Borel's conjecture" and i came across this definition and i need a little bit of help with it's equivalent form.
X has universal measure zero if $f[X]$ has Lebesgue measure zero for each homeomorphism $f:[0, 1] \rightarrow [0, 1]$, equivalently, for each nonatomic, nonnegative, real valued Baire measure $\mu$ on $[0, 1]$, $\mu (X) = 0$.
Now i am stuck here and i think i need to study a bit more of analysis and i would really appreciate any good references related to this paper(atleast the parts related to analysis and measure theory). Also i would be really glad if someone could point me in the right direction with this equivalence. Thanks for your patience.