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In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra"; also σ-field, where the σ comes from the German "Summe" [1]) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections. The ordered pair (,) is called a measurable space.
Greek letters are used in mathematics, science, engineering, and other areas where mathematical notation is used as symbols for constants, special functions, and also conventionally for variables representing certain quantities. In these contexts, the capital letters and the small letters represent distinct and unrelated entities.
Random variables are usually written in upper case Roman letters, such as or and so on. Random variables, in this context, usually refer to something in words, such as "the height of a subject" for a continuous variable, or "the number of cars in the school car park" for a discrete variable, or "the colour of the next bicycle" for a categorical variable.
Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations, and any other mathematical objects and assembling them into expressions and formulas. Mathematical notation is widely used in mathematics , science , and engineering for representing complex concepts and properties in a concise ...
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for ...
μ (1) For a cardinal μ, this is the same cardinal in the next higher type. *104.03 μ (1) For a cardinal μ, this is the same cardinal in the next lower type. *105.03 + The disjoint union of two classes *110.01 + c: The sum of two cardinals *110.02 Crp Short for "correspondence". *110.02 ς (A Greek sigma used at the end of a word.)
is a σ-algebra and = is a filtration. F {\displaystyle \mathbb {F} } really is a filtration, since by definition all F n {\displaystyle {\mathcal {F}}_{n}} are σ -algebras and σ ( X k ∣ k ≤ n ) ⊆ σ ( X k ∣ k ≤ n + 1 ) . {\displaystyle \sigma (X_{k}\mid k\leq n)\subseteq \sigma (X_{k}\mid k\leq n+1).}
Here, is the Hyperbolic tangent and the superscript of -1, above it, is shorthand notation for the inverse function of the Trigonometric function, tangent. For k > 1 {\displaystyle k>1} , a smooth approximation of the sign function is sgn x ≈ tanh k x . {\displaystyle \operatorname {sgn} x\approx \tanh kx\,.}