Let $X = \{(a_1, a_2, . . .)| a_{k} ∈ \{0, 1\} \text{for every k}$ be the set of $\{0, 1\}$-sequences. Let $µ$ be the measure on $X$ for which $µ({(a_1, a_2, . . .) ∈ X | a_1 = b_1, . . . , a_N = b_N }) = \frac{1}{2^N}$ for every $b_1, . . . , b_N ∈ {0, 1}.$ Let $f : X → R$ be defined as $f(a_1, a_2, . . .) := \sum_{k=1}^{\infty}\frac{a_{k}}{2^k}.$
I am trying to calculate the push-forward of measure for every interval $[a,b]$ on the real line,
I noticed that $f(a_1,a_2,.....)$ lies in the $[0,1]$ interval, so for if my interval is $[0,1]$ the push forward is $µ(X)$, but how do I compute measure of $X.$ please assist