This result is essentially the fact that irrational rotations are ergodic. The best way to prove it is by using Fourier series.

Let f be the characteristic function of E,

Extend the domain of f from [0,1] to

by making it periodic with period 1. The Fourier coefficients of f are given by

. The fact that E is invariant under a translation through

*a* tells you that

for all n. But

*a* is irrational, so

can never be equal to 1 unless n=0. Therefore

for all

, which tells you that f is (almost everywhere equal to) a constant function. Since

the constant can only be 0 or 1. Thus m(E) must be 0 or 1.