Check that the set functions defined in Example 4.5 are measures in the sense of Definition 4.1 .

Data From example 4.5

Data from definition 4.1

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(i) (Dirac measure, unit mass) Let (X, A) be a measurable space and let x = X be some point. Then dx: {0, 1}, defined for AE by 5x (A):= is a measure. It is called Dirac's delta measure or unit mass at the point x. (ii) Consider (R, A) with from Example 3.3(v) (i.e. AE if A or A is countable). Then y:{0, 1}, defined for AE by 7(4):= 1 if .x & A, if x 4, 0 if A is countable, 1 otherwise, is a measure. (iii) (Counting measure) Let (X, |4|:= defines a measure. It is called counting measure. (iv) (Discrete probability measure) Let = {w, 2,...} be a countable set and (Pn)neN be a sequence of real numbers p [0, 1] such that EnEN Pn = 1. On (N, P()) the set function are measures. ) be a measurable space and AE. Then #Aif A is finite, +o if A is infinite, P(A) = Pn=, (4), ACN, W:WHEA HEN defines a probability measure. The triplet (N, P(N), P) is called discrete probability space. (v) (Trivial measures) Let (X, A) be a measurable space and AEA. Then (A):= 0 if A = 0, +o if 40 and v(A): 0 Our list of examples does not include the most familiar of all measures: length, area and volume.