Abstract
In finitary combinatorics, one often encounters counting arguments involving the (normalized) counting measure on the finite set under consideration. The continuous analogue of such a basic tool is the Lebesgue measure on [0, 1], [0, 1]n or, more generally, some other probability space. Such an analogy can be made precise through the nonstandard perspective. Indeed, one can consider a hyperfinite set, such as the interval [1, N] = {1, …, N} in \({ }^{\ast }\mathbb {N}\) for some hypernatural number N, endowed with its internal counting measure (defined on the algebra of its internal subsets). Such an internal object in turns gives rise to an (external) probability measure, called Loeb measure. As it turns out, the Lebesgue measure on [0, 1] can be regarded as a restriction of the Loeb measure to a suitable σ-algebra. This makes precise the intuition that the Lebesgue measure is a limit “at infinity” of normalized counting measures. In this chapter, we will present the construction of the Loeb measure and some of its fundamental properties.
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Notes
- 1.
Unfortunately, standard is used in a different sense than in the rest of this book. Indeed, here, a standard probability space is simply a probability space which is isomorphic to a quotient of \(\left [ 0,1\right ] \) endowed with the Borel σ-algebra and Lebesgue measure.
- 2.
Of course, one can use the ergodic theorem to prove the existence of typical elements. However, we need a proof that typical elements exist that does not use the ergodic theorem. One can see, for example, [81, Lemma 2] for such a proof.
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Nasso, M.D., Goldbring, I., Lupini, M. (2019). Loeb Measure. In: Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory. Lecture Notes in Mathematics, vol 2239. Springer, Cham. https://doi.org/10.1007/978-3-030-17956-4_5
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