## Appendix A Measurable Spaces and Related Concepts  
**Measurable space. ** A _measurable space_ is a pair \(\mathcal{S}=(S,\mathcal{A})\), where \(S\) is a set, and \(\mathcal{A}\) is special set system, a \(\sigma\)_-algebra_ over \(S\). The \(\sigma\)-algebra \(\mathcal{A}\) is a family of subsets of \(X\), with the following properties: it contains \(X\) itself, it is closed under taking complements and countable unions. (These properties imply that \(\mathcal{A}\) also contains the empty set and is closed under countable intersections, as well.) If \(S\) is finite or countably infinite, then \(\mathcal{A}\) often simply contains all subsets of \(S\), but this is not necessary. The sets that are contained in \(\mathcal{A}\) are called _measurable sets_. Why do we need to distinguish measurable sets? Because in some situations, typically for non-countable models, we cannot avoid dealing with non-measurable sets (see later, under the heading _Non-measurable sets_). For simplicity, when we talk about a measurable space, we often just denote it by its underlying set \(S\), rather than the more precise \(\mathcal{S}=(S,\mathcal{A})\) notation; this usually does not cause any confusion. **Measurable function. ** If \(\mathcal{S}_{1}=(S_{1},\mathcal{A}_{1})\) and \(\mathcal{S}_{2}=(S_{2},\mathcal{A}_{2})\) are two (not necessarily different) measurable spaces, then a function \(f:S_{1}\mapsto S_{2}\) is called a _measurable function_ if for any \(A\in\mathcal{A}_{2}\), the set of elements that the function maps into \(A\) constitute a measurable set in \(\mathcal{S}_{1}\). That is, \(\{x\,|\,f(x)\in A\}\in\mathcal{A}_{1}\). The set \(f^{-1}(A)=\{x\,|\,f(x)\in A\}\) is called the _pre-image_ of \(A\). Informally, the condition is often expressed this way: the function is measurable if and only if the pre-image of any measurable set is also measurable. **Isomorphic measurable spaces. ** Two measurable spaces \(\mathcal{S}_{1}=(S_{1},\mathcal{A}_{1})\) and \(\mathcal{S}_{2}=(S_{2},\mathcal{A}_{2})\) are called _isomorphic_, if there exists a bijection (1-1 onto function) \(f:S_{1}\mapsto S_{2}\), such that both \(f\) and its inverse are measurable. **Borel measurable space. ** The _Borel subsets_ of **R** (the set of real numbers) are the sets that arise by repeatedly taking countable unions, countable intersections and relative complements (set differences) of open sets. A measurable space is called a _Borel measurable space_ if it is isomorphic to a Borel subset of **R**. **Measure space. ** Note that no measure is included in the definition of a measurable space. If a measure is also added, then it becomes a _measure space_, not to be confused with a _measurable_ space. A _measure_ is a function that assigns a real number to every measurable set, such that certain axioms are satisfied. Specifically, if \(\mathcal{S}=(\mathcal{S},\mathcal{A})\) is a measurable space, then a function \(\mu:\mathcal{A}\mapsto\mathbf{R}\) is a measure, if it is non-negative, \(\mu(\mathcal{O})=0\), and is countably additive. The other means that for every countable collection of sets \(A_{1},A_{2},\ldots\in\mathcal{A}\) it holds that \(\mu(\cup_{i=1}^{\infty}A_{i})=\sum_{i=1}^{\infty}\mu(A_{i})\). Then the triple \(\mathcal{M}=(S,\mathcal{A},\mu)\) is referred to as a measure space. **Probability space. ** A _probability measure_ is a measure with the additional requirement that the measure of the whole space is 1. If this is satisfied, then the arising measure space is referred to as a _probability space_ or a _probability triple_. The parts are often denoted differently from the notation \((S,\mathcal{A},\mu)\) of a general measure space. A frequently used notation for a probability space is \((\Omega,\mathcal{F},P)\), where \(\Omega\) is the set of possible outcomes (elementary events), \(\mathcal{F}\) is the collection of events, and \(P\) is the probability measure. **Non-measurable sets.