Use Lemma 7.2 to show that \(\tau_{x}: y \mapsto y-x, x, y \in \mathbb{R}^{n}\), is \(\mathscr{B}\left(\mathbb{R}^{n}ight) / \mathscr{B}\left(\mathbb{R}^{n}ight)\)-measurable.

Data from lemma 7.2

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Let (X, A), (X', A') be measurable spaces and let A'= o(9'). Then T:XX' is A/A -measurable if, and only if, T- (G) CA, i.e. if T(G)EA VG EG'. (7.2) Proof If T is A/A'-measurable, we have T- (9) CT- (A) CA, and (7.2) is obviously satisfied. Conversely, consider the system := {A' CX': T-(A)EA). By (7.2), 'C' and it is not difficult to see that is itself a o-algebra since T-1 commutes with all set operations. [] Therefore, !! =o(g') co(')=' = T (A') Ed VA'E'. On a topological space (X, 6)- see page 18 - we usually consider the (topo- logical) Borel o-algebra B(x):=o(6). The interplay between measurability and topology can be intricate. One of the simple and extremely useful aspects is the fact that continuous maps are measurable; let us check this for R".