Finite independent Laplace noise has product density

Define $f(z)={\prod }_{i=1}^d{f}_b(z_i)=(2b{)}^{-d}\exp (-\mathrm{\parallel }z{\mathrm{\parallel }}_1\mathrm{/}b)$; the equality factoring the exponential of a finite sum uses [F8]. The function is nonnegative and product-measurable. Repeated application of [F4] gives ${\int }_{{\mathbb{R}}^d}f(z)d{\lambda }^d(z)={\prod }_{i=1}^d{\int }_{\mathbb{R}}{f}_b(x)d\lambda (x)=1$, using [F1] for each one-dimensional integral. By [F5], $\nu (A)={\int }_{A}fd{\lambda }^d$ is therefore a probability measure.

For a measurable rectangle $A={\prod }_{i=1}^d{A}_i$, repeated Tonelli integration [F4] gives $\nu (A)={\prod }_{i=1}^d{\int }_{A_i}{f}_{b}d\lambda ={\prod }_{i=1}^d\mathrm{Lap}(b)(A_i)$. The finite product law has this same rectangle value by [F3]. Rectangles form a generating pi-system by [F2], and the constant sequence of whole-space rectangles is an increasing exhaustion with mass one for both measures, so [F6] identifies $\nu$ with $\mathrm{Lap}(b{)}^{\otimes d}$ on the entire product sigma-algebra. Taking unrestricted factors equal to $\mathbb{R}$ gives the stated coordinate marginals and factorization for every finite subfamily of coordinate events. These are exactly the events in the generated coordinate sigma-algebras, so the projections are independent by [F7].