Let
x − ai T(x) = ai+1 − ai
with x ∈ [ai,ai+1). T : J → J where J = [0,1). And 0 = a0 < a1 < ··· < ak = 1. We want to show that T is a measure preserving transformation of (J,L(J),λ) regardless of choice of {ai}.
We have that T−1(x) = x(ai+1−ai)+ai. Let C be the collection of left-closed, right-open dyadic intervals in [0,1). We saw in Section 2.7 that C is a sufficient semi-ring. For I we write I = [k/2i,(k + 1)/2i) for integers k,i with i ≥ 0 and k ∈ {0,...,2i − 1}. Observe that λ(I) = 1/2i for all I ∈ C. Assume for a fixed I that I ⊂ [ai,ai+1) for some i ∈ {0,...k}.
Then,
" !
−1 k k + 1
T (I) = 2 i(ai+1 − ai) + ai, i (ai+1 − ai) + ai
2
T−1(I) is an interval for any I and is hence a measurable set. Moreover,
−1 k + 1 k
λ(T (I)) = i (ai+1 − ai) + ai − ( i(ai+1 − ai) + ai)
2 2
k + 1 k + 1 k k
= 2i ai+1 − 2i ai − 2iai+1 + 2iai
(k + 1)(ai+1 − ai) − k(ai+1 − ai)
=
2i
((k + 1) − k)(ai+1 − ai)
=
=
2i
Observe that T maps some values onto I for each T defined on the intervals [ai,ai+1). Hence, there will be k such intervals resulting from I with the same length as the above when applying T−1.
1
If we add this up over all intervals [ai,ai+1), we get,
ak − a0 1
= = λ(I) 2i 2i
as required. Hence, we can apply Theorem 3.4.1 in order to assert that T is then a measure-preserving transformation.
