Exercise 14, § 8

Let $K$ be the interval $[0,1]$ of $\mathbf{R}$, $\lambda$ the Lebesgue measure on $K$, $A$ an uncountable set, and $\mu$ the measure on $T = K^A$ that is the product of the measure $\lambda$ on each of the factors.

a) Show that every closed and metrizable subspace X of T is $\mu$-negligible (make use of Exer. 13 b) above, and Exer. 8 of GT, IX, §2).

b) Let Y be a locally compact space admitting a countable base, $\nu$ a positive measure on Y, and $\pi$ a $\nu$-proper mapping of Y into T. Show that the image $\pi(\nu)$ is alien to $\mu$ (using a), show that $\pi(\nu)$ is concentrated on a $\mu$-negligible set).