# Using strong law of large numbers to construct a measure

How can I apply the strong law of large numbers to construct a measure $$\{\mu_p\}$$ on $$([0,1],\mathcal{B}([0,1]))$$ such that $${\mu_p}$$'s are singular to $$\lambda_{Leb}$$ and the distribution function of $$\mu_p$$ is continuous?

To be more precise for such $$\mu_p$$, if $$p_1 \neq p_2$$, then $$\mu_{p_1} \perp \mu_{p_2}$$, and in particular if $$p \neq \frac{1}{2}$$, $$\mu_p \perp \lambda$$ where $$\lambda$$ is the Lebesgue measure on $$\mathbb{R}$$.