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We establish global universal approximation theorems on spaces of piecewise linear paths, stating that linear functionals of the corresponding signatures are dense with respect to $L^p$- and weighted norms, under an integrability condition on the underlying weight function. As an application, we show that piecewise linear interpolations of Brownian motion satisfies this integrability condition. Consequently, we obtain $L^p$-approximation results for path-dependent functionals, random ordinary differential equations, and stochastic differential equations driven by Brownian motion.