The two dimensions to measure are the length and width of the area you need to calculate. M+(X × Y, A ⊗ B) to M+(X × Y), with the product sigma-field assumed, or to M+(A⊗ B), with the product space assumed. This yields a product called the area, which is expressed in square feet (or square inches if you are calculating a much smaller space, such as a dollhouse). The category of measurable spaces consists of objects ( X, B X) and measurable morphisms ϕ: ( X, B X) → ( Y, B Y). To find square feet in a room, first measure the dimensions of your space. Suppose that (X;A) and (Y;B) are measurable spaces. These two measurements should be equal, but if they are not, work with the shorter of the two measurements. De nition 5.1. 9 Expectation 16 10 Conditional expectation and probability 17 11 Conditional probability 21 ... is called a measure space. 2 A measure on (S;S) consists of a nonempty subset, M, of S, together with a mapping M ! Similarly, Mbdd(A⊗ B) will be an abbreviation for Mbdd(X×Y, A⊗B), the vector space of all bounded, real-valued, product measurable functions on X×Y. measure) to create a product measure on a product space. Nonzero finite measures are analogous to probability measures in the sense that any finite measure μ is proportional to the probability measure $${\displaystyle {\frac {1}{\mu (X)}}\mu }$$. Product Measures 7.1 The Product Measure Theorem Problem 7.1.1. A measure μ is called σ-finite if X can be decomposed into a countable union of measurable sets of finite measure. R+ (where R+ 2In more detail: Set S1 = P. Let S2 the set that results from applying the above process to S1; then by S3 the set that results from applying the process to S2, etc. Using a tape measure, check out the height from top to bottom and the width from side to side. The product of a finite or countable family of countably generated measurable spaces is countably generated. Conceptually, this is similar to defining the Cartesian product of sets and the product topology of two topological spaces, except that there can be many natural choices for the product measure. Suppose that EˆX Y. In mathematics, given two measurable spaces and measures on them, one can obtain a product measurable space and a product measure on that space. To find square feet, multiply the length measurement in feet by the width measurement in feet.

They coincide with the standard product ( X × Y, B X × B Y), where X × Y is the Cartesian product of X and Y and B X × B Y is the coarsest σ -algebra on X × Y such that the canonical projections π X: X × Y → X and π Y: X × Y → Y … Measure All Sides of the Space. Fix a tape measure or other measuring tool to one end of the length and extend it to the other end. The prod- uct ˙-algebra A Bis the ˙-algebra on X Y generated by the collection of all measurable rectangles, A B= ˙(fA B: A2A, B2Bg): The product of (X;A) and (Y;B) is the measurable space (X Y;A B). Let (X;A) and (Y;B) be measurable spaces.