ABSTRACT
If (Ω, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315180700/dd402744-6e80-442b-b0e2-8102d4d6e607/content/c004_iequ_0001.tif"/> A , P) is a measure space as defined in the previous chapter and the measure P assigns 1 to the set Ω (i.e., P(Ω) = 1), then P is called a probability measure or sometimes just a probability, and the triple (Ω, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315180700/dd402744-6e80-442b-b0e2-8102d4d6e607/content/c004_iequ_0002.tif"/> A , P) is then called a probability space. Because P(Ω) = 1, P assigns a number no greater than 1 to any set in the sigma-algebra https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315180700/dd402744-6e80-442b-b0e2-8102d4d6e607/content/c004_iequ_0003.tif"/> A , and the sum of probabilities assigned by P to the sets of a partition must sum to 1 across the partition, which is just what we expect from a probability. Note that for any measure space (Ω, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315180700/dd402744-6e80-442b-b0e2-8102d4d6e607/content/c004_iequ_0004.tif"/> A , µ) with µ(Ω) finite and nonzero, a new measure can be defined by dividing µ’s assignment for any set by the number that µ assigns to Ω. That is, P(A) = µ(A)/µ(Ω) for all sets A in the sigma-algebra https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315180700/dd402744-6e80-442b-b0e2-8102d4d6e607/content/c004_iequ_0005.tif"/> A is a probability measure on (Ω, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315180700/dd402744-6e80-442b-b0e2-8102d4d6e607/content/c004_iequ_0006.tif"/> A ).
