The cumulative distribution function of singular measure has a.e. 0 derivative. [on hold]

Let μ\mu be a finite measure in Rn\mathbb R^n singular to Lebesgue measure i.e. ∃E⊂Rn,|E|=0,μ(E)=μ(Rn)\exists E\subset\mathbb R^n,|E|=0,\mu(E)=\mu(\mathbb R^n).

Consider the cumulative distribution function F(x1,…,xn)=μ((−∞,x1)×⋯×(−∞,xn))F(x_1,…,x_n)=\mu((-\infty,x_1)\times\cdots\times(-\infty,x_n)), show that for almost everywhere x∈Rnx\in\mathbb R^n w.r.t Leb measure, dFdxi(x)\frac{dF}{dx_i}(x) exists and is equal to zero.

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