gromit1987

Can someone explain the meaning of the sentence: ``All distributions in this text are finitely supported over the σ-algebra of all subsets of X'' ? On the one hand, it is impossible to define a probability measure on the σ-algebra of all subsets of [0,1]. On the other hand, if a distribution is finitely supported, then there is no point in using a σ-algebra different than what is generated by its support, right?

When dealing with bits, the situation is different since algebraic degrees of freedom (dimensions or coefficients of sinusoids) are different than information degrees of freedom (bits). This difference in the context of the sampling theorem is explored in https://arxiv.org/abs/1601.06421, where it is shown that sampling below the Nyquist rate (without additional loss) is possible when the samples must be quantized to satisfy some bit (information) constraint.