\text { Let } q \in(0,1) \text {. Prove that } \nu=\sum_{k=0}^{\infty}(1-q) q^{k} \delta_{k} \text { is a probability measure on }(\mathbb{R}, \mathscr{B}(\mathbb{R})) \text {. Here } \delta_{k} \text {

denotes the Dirac measure } with mass at k. (ii) Let q = 1/2 in the definition of v above. Find: (а) v([-1,1); (b) v((0, 1)); v(N), where N = {1, 2, 3, . .}.