A field of sets is a collection or system of sets, where sum, product and difference of any two sets, forms another set in the same field. Thus a field of sets is characterised by closure under standard set operations.
Friday 21 February 2020
Mathematical Analogies That Made Probability Theory A Reality
Probability, likelihood, chance - these concepts have an intuitive existence - linked to our perception of the "odds" of particular events.
But in today's world, probability also has a formal definition to go with its intuitive meaning, and this definition was an axiomatic one, spelled out by Kolmogorov. He achieved this by appealing to Lebesgue's theories of measure and integration. These ideas are linked in the form of the Lebesgue integral. Riemann's ideas alone are not enough to support the axiomatisation of the probability concept.
The relevant ideas and analogies are as follows.
Events as sets
Measure of a set -> Probability of an event
Integral of a function -> Mathematical expectation of a random variable
Analogies led fortuitously to further analogies; for example independence of random variables paralleled harmoniously with the corresponding properties of orthogonal functions.
Quite fundamental to Kolmogorov's axiomatisation is the idea of a field of sets (not to be confused with fields in ring theory e.g. field of real numbers, rationals etc.), where the field of interest, so-to-speak, is the set of events which are (physically) subsets of elementary events. It is denoted Curly-F. The elements of this set, Curly-F, are called random events.
The idea of a field of sets is defined in Hausdorff's Mengenlehre (a German idiom for set theory, with Mengen literally meaning quantity and lehre meaning teaching). Hausdorff was known for his involvement in topology.
But in today's world, probability also has a formal definition to go with its intuitive meaning, and this definition was an axiomatic one, spelled out by Kolmogorov. He achieved this by appealing to Lebesgue's theories of measure and integration. These ideas are linked in the form of the Lebesgue integral. Riemann's ideas alone are not enough to support the axiomatisation of the probability concept.
The relevant ideas and analogies are as follows.
Events as sets
Measure of a set -> Probability of an event
Integral of a function -> Mathematical expectation of a random variable
Analogies led fortuitously to further analogies; for example independence of random variables paralleled harmoniously with the corresponding properties of orthogonal functions.
Quite fundamental to Kolmogorov's axiomatisation is the idea of a field of sets (not to be confused with fields in ring theory e.g. field of real numbers, rationals etc.), where the field of interest, so-to-speak, is the set of events which are (physically) subsets of elementary events. It is denoted Curly-F. The elements of this set, Curly-F, are called random events.
The idea of a field of sets is defined in Hausdorff's Mengenlehre (a German idiom for set theory, with Mengen literally meaning quantity and lehre meaning teaching). Hausdorff was known for his involvement in topology.
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