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We show that standard feedforward networks with as few as a single hidden layer can uniformly approximate continuous functions on compacta provided that the activation function @j is locally Riemann integrable and nonpolynomial, and have universal L^p (@m) approximation capabilities for finite and compactly supported input environment measures @m provided that @j is locally bounded and nonpolynomial. In both cases, the input-to-hidden weights and hidden layer biases can be constrained to arbitrarily small sets; if in addition @j is locally analytic a single universal bias will do.