Define $T_p: \mathbb{T}\longrightarrow\mathbb{T}$ to be $T_p(x)=px \ mod\mathbb{Z}$, Furstenberg x2 x3 conjecture is as follows:
The unique non-atomic (ergodic) borel measure on $\mathbb{T}$ that is $T_2$ and $T_3$ invariant is Lebesgue measure.
In some sources I saw the measure was required to be ergodic while in others ergodicity was omitted. In this MO question ergodicity is omitted and it is argued in the comments that the two formulations are equivalent but I couldn't understand why. To be precise, my question is why Furstenberg "ergodic" conjecture implies Furstenberg "non-ergodic" conjecture, or in what sense these conjectures are equivalent.