Theorem 4.2.8 in Dudley's Real Analysis and Probability states:
Given a set $X$, a measurable space ($Y$, $\mathcal{B}$), and a function $T$ from $X$ into $Y$ , a real-valued function $f$ on $X$ is $T^{−1}[\mathcal{B}]$ measurable on $X$ if and only if $f = g \circ T$ for some $\mathcal{B}$-measurable function $g$ on $Y$ .
And then here is problem 4.2.6:
In Theorem 4.2.8, let $X = \mathbb{R}$, let $Y$ be the unit circle in $\mathbb{R}^2$: $Y := \{(x, y): x^2 + y^2 = 1\}$, and $\mathcal{B}$ the Borel $\sigma$-algebra on $Y$ . Let $T (u) := (\cos u,\sin u)$ for all $u \in \mathbb{R}$. Find which of the following functions $f$ on $\mathbb{R}$ are $T^{−1}[\mathcal{B}]$ measurable, and for those that are, find a function $g$ as in Theorem 4.2.8:
(a) $f (t) = \cos(2t)$; (b) $f (t) = \sin(t/2)$; (c) $f (t) = \sin^2 (t/2)$.
So far, (a) and (c) are $T^{−1}[\mathcal{B}]$measurable, as I can have $g = x^2 -y^2$ and $g = \frac{1-x}{2}$ respectively, and as they are continuous, they are $\mathcal{B}$-measurable (Am I right?).
I also think (b) is not $T^{−1}[\mathcal{B}]$measurable, but I'm not sure how I can show that...
Any help would be greatly appreciated. Thank you.