(a) Let $(E, \mathcal{E}, \mu)$ be a measure space. What does it mean to say that $T: E \rightarrow E$ is a measure-preserving transformation? What does it mean to say that a set $A \in \mathcal{E}$ is invariant under $T$ ? Show that the class of invariant sets forms a $\sigma$-algebra.

(b) Take $E$ to be $[0,1)$ with Lebesgue measure on its Borel $\sigma$-algebra. Show that the baker's map $T:[0,1) \rightarrow[0,1)$ defined by

$$T(x)=2 x-\lfloor 2 x\rfloor$$

is measure-preserving.

(c) Describe in detail the construction of the canonical model for sequences of independent random variables having a given distribution $m$.

Define the Bernoulli shift map and prove it is a measure-preserving ergodic transformation.

[You may use without proof other results concerning sequences of independent random variables proved in the course, provided you state these clearly.]