I'm trying to understand the proof given by D. Rudolph in his paper "x2 and x3 invariant measures and entropy". I'm particularly trying to undestand the proof of lema 4.4.

Let's consider a secuence of probability measures $\delta(\hat{y},n)$ which concentrates on $\{\frac{0}{p^n}, \dots, \frac{p^n-1}{p^n}\}$ for each $n \in \mathbb{N}$. One shows that each of the measures $\delta(\hat{y},n)$ is invariant under a shift (mod 1) by a number $a_n$ which is a fraction, in least terms, with denominator $\geq 2^{n-i_0+1}$ where $i_0 \geq 0$ is a fixed index. Then the paper states:

" Thus the group of shifts (mod 1) preserving $\delta(\hat{y},n)$ is of order at least $2^{n - i_0+1}$, and its minimal element $d_n \leq \frac{1}{2^{n-i_0+1}}$. For any continuous function $f$ on $[0,1), f(0) = 0, f(1) = 1$, this forces $$ \lim_{n \to \infty} \int f \ d\delta(\hat{y},n) = \int f dm $$ and hence $\delta(\hat{y},n)$ converges weakly to $m$ the Lebesgue measure."

I'm having trouble understanding the fragment exposed before. Particularly that does it mean that the group of shifts preserveving the measure is of certain **order**? What is a **minimal element**? How does this force the stated convergence?

Any help is aprecciated