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37697 results in Cambridge Textbooks

Page 1853 of 1885

  • Chapter 1 - Measure Spaces

  • from PART A - FOUNDATIONS
  • David Williams, Statistical Laboratory, University of Cambridge
  • Book:
    Probability with Martingales
    Published online:
    05 June 2012
    Print publication:
    14 February 1991, pp 14-22

  • Summary

    Introductory remarks

    Topology is about open sets. The characterizing property of a continuous function f is that the inverse image f−1(G) of an open set G is open.

    Measure theory is about measurable sets. The characterizing property of a measurable function f is that the inverse image f−1(A) of any measurable set is measurable.

    In topology, one axiomatizes the notion of ‘open set’, insisting in particular that the union of any collection of open sets is open, and that the intersection of a finite collection of open sets is open.

    In measure theory, one axiomatizes the notion of ‘measurable set’, insisting that the union of a countable collection of measurable sets is measurable, and that the intersection of a countable collection of measurable sets is also measurable. Also, the complement of a measurable set must be measurable, and the whole space must be measurable. Thus the measurable sets form a σ-algebra, a structure stable (or ‘closed’) under countably many set operations. Without the insistence that ‘only countably many operations are allowed’, measure theory would be self-contradictory – a point lost on certain philosophers of probability.

  • Chapter E - Exercises

  • from APPENDICES
  • David Williams, Statistical Laboratory, University of Cambridge
  • Book:
    Probability with Martingales
    Published online:
    05 June 2012
    Print publication:
    14 February 1991, pp 224-242

  • Summary

    Starred exercises are more tricky. The first number in an exercise gives a rough indication of which chapter it depends on. ‘G’ stands for ‘a bit of gumption is all that's necessary’. A number of exercises may also be found in the main text. Some are repeated here. We begin with an

    Antidote to measure-theoretic material – just for fun, though the point that probability is more than mere measure theory needs hammering home.

    EG.1. Two points are chosen at random on a line AB, each point being chosen according to the uniform distribution on AB, and the choices being made independently of each other. The line AB may now be regarded as divided into three parts. What is the probability that they may be made into a triangle?

    EG.2. Planet X is a ball with centre O. Three spaceships A, B and C land at random on its surface, their positions being independent and each uniformly distributed on the surface. Spaceships A and B can communicate directly by radio if ∡AOB < 90°. Show that the probability that they can keep in touch (with, for example, A communicating with B via C if necessary) is (π + 2)/(4π).

    EG.3. Let G be the free group with two generators a and b. Start at time 0 with the unit element 1, the empty word.

Page 1853 of 1885