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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 $\mathbb{R}$.

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