These pages are from a book in preparation: Mapping of Probabilities Theory for the Interpretation of Uncertain Physical Measurements by Albert Tarantola (to be submitted to Cambridge University Press) The aim of the book is to develop the mathematical bases necessary for a proper treatment of measurement uncertainties (and this includes the formulation of Inverse Problems). I don’t think that the right setting for this kind of problems is the usual one (based on conditional probabilities). This is why I develop here some new notions: image of a probability, reciprocal image of a probability, intersection of probabilities [these are generalizations of the usual operations on sets]). This text is still confidential: you can read it and perhaps, learn some new things, but you are not allowed to publish results based on the notions presented in this text, unless you ask me ([email protected]) for a permission. Please send me any comment you may have. Chapter 1 Sets This is the introduction to the chapter. This is the introduction to the chapter. This is the introduction to the chapter. This is the introduction to the chapter. This is the introduction to the chapter. This is the introduction to the chapter. This is the intro- duction to the chapter. This is the introduction to the chapter. This is the introduction to the chapter. This is the introduction to the chapter. 2 Sets 1.1 Sets 1.1.1 Relations The elements of a mathematical theory may have some properties, and may have some mutual relations. The elements are denoted by symbols (like x or 2 ), and the relations are denoted by inserting other symbols between the elements (like = or ∈ ). The element denoted by a symbol may be variable or may be determined. Given an initial set of (non contradictory) relations, other relations may be demon- strated to be true or false. A relation (or property) containing variable elements is an identity if it becomes true for any determined value given to the variables. If R and S are two relations containing variable elements, one says that R implies S , and one writes R ⇒ S , if S is true every time that R is true. If R ⇒ S and S ⇒ R , then one writes R ⇔ S , and one says that R and S are equivalent (or that R is true if, and only if, S is true). The relation ¬R , the negation of R , is true if R is false. Therefore, one has ¬( ¬(R)) ⇔ R . (1.1) From R ⇒ S it follows ¬S ⇒ ¬R : ( R ⇒ S ) ⇒ ((¬S) ⇒ (¬R)) . (1.2) (but it does not follow (¬R) ⇒ (¬S) ). If R and S are two relations, then, R OR S is also a relation, that is true if at least one of the two relations {R , S} is true. Similarly, the relation R AND S is true only if the two relations {R , S} are both true. Therefore, for any two relations {R , S} , the relation R OR S is false only if both R and S are false: ¬( R OR S ) ⇔ (¬R) AND (¬S) , (1.3) and R AND S is false if any of the R , S is false: ¬( R AND S ) ⇔ (¬R) OR (¬S) . (1.4) In theories relations like a = b and a ∈ A make sense, the relation ¬(a = b) is written a 6= b , while the relation ¬(a ∈ A) is written a ∈/ A. 1.1 Sets 3 1.1.2 Sets A set is a “well-defined collection” of (abstract) elements. An element belongs to a set, or is a member of a set. If an element a is member of a set A one writes a ∈ A (or a ∈/ A if a is not member of A ). If a and b are elements of a given set, they may be different elements ( a 6= b ), or they may, in fact, be the same element ( a = b ). Two sets A and B are equal if they have the same elements, and one writes A = B (or A 6= B if they are different). If a set A consists of the elements a, b, . , one writes A = {a, b,... } . The empty set, the set without any element, is denoted ∅ . The set of all the subsets of a set A is called the power set of A , and is denoted ℘[A] . The Cartesian product of two sets A and B , denoted A × B , is the set whose elements are of all the ordered pairs (a, b) , with a ∈ A and b ∈ B. Given two sets A and B one says that A is a subset of B if every member of A is also a member of B . One then writes A ⊆ B , and one also says that A is contained in B . In particular, any set A is a subset of itself, A ⊆ A . If A ⊆ B but A 6= B one says that A is a proper subset of B , and one writes A ⊂ B. The union of two sets A1 and A2 , denoted A1 ∪ A2 , is the set consisting of all elements that belong to A1 , or to A2 , or to both: a ∈ A1 ∪ A2 ⇔ a ∈ A1 OR a ∈ A2 . (1.5) The intersection of two sets A1 and A2 , denoted A1 ∩ A2 , is the set consisting of all elements that belong to both A1 and A2 : a ∈ A1 ∩ A2 ⇔ a ∈ A1 AND a ∈ A2 . (1.6) If A1 ∩ A2 = ∅ , one says that A1 and A2 are disjoint. Given some reference set A0 , to any subset A of A0 , one associates its complement, that is the set of all the elements of A0 that are not members of A . The complement of a set A (with respect to some reference set) is denoted Ac . 1 Given some reference set A0 , the indicator function of a set A ⊆ A0 is the func- tion that to every element a ∈ A0 associates the number one, if a ∈ A , or the number zero, if a ∈/ A (see figure 1.1). This function may be denoted by a symbol like χA or ξA . For instance, using the former, ( 1 if a ∈ A χA(a) = (1.7) 0 if a ∈/ A. The union and intersection of sets can be expressed in term of indicator functions as χA ∪ A = max{χA , χA } = χA + χA − χA χA 1 2 1 2 1 2 1 2 (1.8) χ χ χ χ χ A1 ∩ A2 = min{ A1 , A2 } = A1 A2 . 1The indicator function is sometimes called characteristic function, but there is another sense for that name in probability theory. 4 Sets 0 A0 0 A0 A a 1 1 1 1 0 0 a 1 0 A Figure 1.1: The indicator of a subset A (of a given set) is the function that takes the value one for every element of the subset, and the value zero for the elements out of the subset. As two subsets are equal if their indicators are equal, the properties of the two oper- ations ∪ and ∩ (like those in equations 1.10–1.12) can be demonstrated using the numerical relations 1.8. A partition of a set A is a set P of subsets of A such that the union of all the subsets equals A (the elements of P “cover” A ) and such that the intersection of any two subsets is empty (the elements are “pairwise disjoint”). The elements of P are called the blocks of the partition. 1.1 Sets 5 1.1.3 Basic Properties Let A1 ,A2 , and A3 be arbitrary subsets of some reference set A0 . From A1 ⊆ A2 and A2 ⊆ A3 it follows A1 ⊆ A3 , while from A1 ⊆ A2 and A2 ⊆ A1 it follows A1 = A2 . Among the many other properties valid, let us remark the De Morgan laws (A ∩ B)c = Ac ∪ Bc ; (A ∪ B)c = Ac ∩ Bc , (1.9) the commutativity relations A1 ∪ A2 = A2 ∪ A1 ;A1 ∩ A2 = A2 ∩ A1 , (1.10) the associativity relations A1 ∪ ( A2 ∪ A3 ) = ( A1 ∪ A2 ) ∪ A3 (1.11) A1 ∩ ( A2 ∩ A3 ) = ( A1 ∩ A2 ) ∩ A3 , and the distributivity relations A1 ∪ ( A2 ∩ A3 ) = ( A1 ∪ A2 ) ∩ ( A1 ∪ A3 ) (1.12) A1 ∩ ( A2 ∪ A3 ) = ( A1 ∩ A2 ) ∪ ( A1 ∩ A3 ) . 6 Sets 1.2 Sigma-Fields 1.2.1 Cardinality of a Set When a set A has finite number of elements, its cardinality, denoted |A| , is the num- ber of elements in the set. Two sets with an infinite number of elements have the same cardinality if the elements of the two sets can be put in a one-to-one correspon- dence (through a bijection). The sets than can be put in correspondence with the set N of natural numbers are called countable (or enumerable). The (infinite) cardinality of N is denoted |N| = ℵ0 (aleph-zero), so if a set A is countable, its cardinality is |A| = ℵ0 . Cantor (1884) proved that the set < of real numbers is not countable: (the real numbers form a continuous set). The (infinite) cardinality of < is denoted |<| = ℵ1 (aleph-one). Any set A that can be put in correspondence with the set of real numbers has, therefore, the cardinality |A| = ℵ1 . One can give a clear sense to the relation ℵ1 > ℵ0 , but we don’t need those details in this book. 1.2 Sigma-Fields 7 1.2.2 Field The power set of a set Ω , denoted ℘(Ω) , has been defined as the set of all possible subsets of Ω . If the set Ω has a finite or a countably infinite number of elements, we can build probability theory on ℘(Ω) , and we can then talk about the probability of any subset of Ω .