Is the set function \(\gamma\) of Example 4.5 (ii) still a measure on the measurable space \((\mathbb{R}, \mathscr{B}(\mathbb{R}))\) ? Is it still a measure on the measurable space \((\mathbb{Q}, \mathbb{Q} \cap \mathscr{B}(\mathbb{R}))\) ?

Data from example 4.5

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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 8x (4) := 1 if .x & A, if x 4, 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):= 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 N = {w, 2,...} be a countable set and (Pn)nN be a sequence of real numbers p [0,1] such that EEN Pn = 1. On (N,P(n)) the set function P(A) = ) be a measurable space and A . Then #Aif A is finite, +o if A is infinite. Pn-Pnown (A), HEN ACQ, H:WHEA defines a probability measure. The triplet (N, P(), P) is called discrete probability space. (v) (Trivial measures) Let (X, A) be a measurable space and A A. Then 0 if A = 0, (4):= + if A0 and v(A):=0 are measures. Our list of examples does not include the most familiar of all measures: length, area and volume.