Analyzing errors and uncertainties in measurement problems

A practical measurement always raises the question of how well it reproduces the concept to be quantified. We present a mathematical perspective on the measurement problem by introducing a conceptual framework that allows an analytical discussion of this question. Within this framework, we define the measurement result, the concept to be measured, the epistemic errors and aleatory uncertainties. We discuss the first quantifications inside a measurement procedure by analyzing the mathematical spaces and operators. Finally, we apply our considerations to voltage measurements with sampling oscilloscopes.