Normal numbers

Imagine if we sample numbers from the unit interval and count the occurrences of each digit in every number’s infinite decimal representation. For instance, in the number , every non-zero digit appears exactly once, but appears infinitely often (trailing zeros). On the other hand, for the number , the digit occurrences (after the “dot”) is equal for every digit. As the number of samples increases, one may expect that the number of occurrences for each digit in to be close to one another. In fact, the Normal Number Theorem, first formulated by Borel, not only formalizes this intuition but goes a step further: for almost every number in , each digit appears […]

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Littlewood’s Three Principles

In measure theory, the notion of measurability restricts sets and functions so that limit operations are sensible. With non-measurabilty manifesting only through the Axiom of Choice, measurability is a semantically rich class, and in particular, Littlewood’s Three Principles specify precisely how measurability relates back to more elementary building blocks as follows: Every measurable set is nearly open or closed. Every measurable function is nearly continuous. Every convergent sequence of measurable functions is nearly uniformly convergent. Here in this post, we flesh out these principles in detail, with an emphasis on how these various concepts approximate one another. Each section corresponds to one principle and is independent from another. We primarily focus on […]

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