From https://www.staff.science.uu.nl/~kraai101/LectureNotesMM-2.pdf
example 1.8.4 (multiplication by 2 modulo 1), page 29.
What do they do at the second equality $\lambda(A\cap T^{-n} B) = \frac{1}{\lambda(A)}\lambda(T^n A\cap B)$?
I'm sure that they don't use the fact that $T^n(B)=[0,1)$ because that information is already used in the third equality.
They apply $T$ on $A$ and $T^{n}(B)$, this $n$ times, so $T$ looks like making any set $\lambda$-twice as big. But this is easily seen not to be true, take $[0,1)$.
I hope anyone can answer this, thanks in advance.