ABSTRACT. Let (X, 3, v) be an internal measure space in a denumerably comprehensive enlargement.The set X is a standard measure space when equipped with the smallest standard o-algebra % containing the algebra a, where the extended real-valued measure p. on % is generated by the standard part of v. If / is fl-measurable, then its standard part / is jR-measurable on X. If / and p. are finite, then the vintegral of / is infinitely close to the /¿-integral of /. Applications include coin tossing and Poisson processes.In particular, there is an elementary proof of the strong Markov property for the stopping time of the ;'th event and a construction of standardsample functions for Poisson processes.

ABSTRACT. Let (X, 3, v) be an internal measure space in a denumerably comprehensive enlargement.The set X is a standard measure space when equipped with the smallest standard o-algebra % containing the algebra a, where the extended real-valued measure p. on % is generated by the standard part of v. If / is fl-measurable, then its standard part / is jR-measurable on X. If / and p. are finite, then the vintegral of / is infinitely close to the /¿-integral of /. Applications include coin tossing and Poisson processes.In particular, there is an elementary proof of the strong Markov property for the stopping time of the ;'th event and a construction of standardsample functions for Poisson processes.

Abstract. This note shows that in terms of known proofs of the Besicovitch Covering Theorem, the best constant for that theorem is the maximum number of points that can be packed into a closed ball of radius 2 when the distance between pairs of points is at least 1 and one of the points is at the center of the ball. Exponential upper and lower bounds are also established.

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