162 1. Measure theory cardinality of the continuum. (The last part of this exercise is only suitable for students who are comfortable with cardinal arithmetic.) Exercise 1.7.19. (i) Show that the product of two trivial σ-algebras (on two different spaces X, Y ) is again trivial. (ii) Show that the product of two atomic σ-algebras is again atomic. (iii) Show that the product of two finite σ-algebras is again finite. (iv) Show that the product of two Borel σ-algebras (on two Euclidean spaces Rd, Rd with d, d 1) is again the Borel σ-algebra (on Rd × Rd Rd+d ). (v) Show that the product of two Lebesgue σ-algebras (on two Eu- clidean spaces Rd, Rd with d, d 1) is not the Lebesgue σ-algebra. (Hint: Argue by contradiction and use Exercise 1.7.18(iii).) (vi) However, show that the Lebesgue σ-algebra on Rd+d is the comple- tion (see Exercise 1.4.26) of the product of the Lebesgue σ-algebras of Rd and Rd with respect to d+d -dimensional Lebesgue measure. (vii) This part of the exercise is only for students who are comfortable with cardinal arithmetic. Give an example to show that the product of two discrete σ-algebras is not necessarily discrete. (viii) On the other hand, show that the product of two discrete σ-algebras 2X, 2Y is again a discrete σ-algebra if at least one of the domains X, Y is at most countably infinite. Now suppose we have two measure spaces (X, BX,μX) and (Y, BY , μY ). Given that we can multiply together the sets X and Y to form a product set X × Y , and can multiply the σ-algebras BX and BY together to form a product σ-algebra BX ×BY , it is natural to expect that we can multiply the two measures μX : BX [0, +∞] and μY : BY [0, +∞] to form a product measure μX × μY : BX × BY [0, +∞]. In view of the “base times height formula” that one learns in elementary school, one expects to have (1.36) μX × μY (E × F ) = μX(E)μY (F ) whenever E BX and F BY . To construct this measure, it is convenient to make the assumption that both spaces are σ-finite: Definition 1.7.10 (σ-finite). A measure space (X, B,μ) is σ-finite if X can be expressed as the countable union of sets of finite measure. Thus, for instance, Rd with Lebesgue measure is σ-finite, as Rd can be expressed as the union of (for instance) the balls B(0,n) for n = 1, 2, 3,...,
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