Exercise 13, § 3

Let $r$ be a number such that $0 < r < 1$; for every integer $n \geq 1$, denote by $\lambda_{n,r}$ the measure $(\varepsilon_{r^n} + \varepsilon_{-r^n})/2$ on $\mathbf{R}$, and set $\mu_{n,r} = \lambda_{1,r} * \lambda_{2,r} * \cdots * \lambda_{n,r}$.

a) Show that the sequence $(\mu_{n,r})$ converges vaguely to a measure $\mu_r$ on $\mathbf{R}$, with support contained in $I = [-1, +1]$ (prove that for every interval U of $\mathbf{R}$, the sequence $(\mu_{n,r}(U))$ is convergent).

b) Show that for $r < 1/2$, the measure $\mu_r$ is alien to the Lebesgue measure on $\mathbf{R}$, but that $\mu_{1/2}$ is the measure induced on I by Lebesgue measure.

c) Let $\nu_{1/4}$ be the image of $\mu_{1/4}$ under the homothety $t \mapsto 2t$ in $\mathbf{R}$. Show that $\mu_{1/4} * \nu_{1/4} = \mu_{1/2}$, even though $\mu_{1/4}$ and $\nu_{1/4}$ are both alien to Lebesgue measure (use Exer. 11 a)).