I'm a little confused about the statement "μ(A)=|ϕ−1(A)|=|ψ(A)| for every A=(0,x)"; don't we have μ(A)=ϕ(x) =|ϕ(A)| by definition of ϕ? Nonetheless, I think this construction of W is correct, so thank you, Mateusz! A simple example of a singular, atomless, full-support probability measure on [0,1] is obtained by i.i.d. drawings of bits from {0,1} with probabilities 2/3 and 1/3, respectively, to obtain the binary expansion of a real number. Then V could be the set of reals whose binary expansions contain, in the limit, twice as many zeroes as ones.

Thanks for sending! This would work, except that S is stipulated to be a closed set.