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Let X be a measurable space, let μ be a measure on X, and let N be a measurable set in X.
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Let and X
" Let X be the unit Cartesian square ×, and let ~ be the equivalence relation on X defined by ∀ a, b ∈ (( a, 0 ) ~ ( a, 1 ) ∧ ( 0, b ) ~ ( 1, b )).
Let ( X < sub > i </ sub >, f < sub > ij </ sub >) be an inverse system of objects and morphisms in a category C ( same definition as above ).
* Let the index set I of an inverse system ( X < sub > i </ sub >, f < sub > ij </ sub >) have a greatest element m. Then the natural projection π < sub > m </ sub >: X → X < sub > m </ sub > is an isomorphism.
Let and be
Let every policeman and park guard keep his eye on John and Jane Doe, lest one piece of bread be placed undetected and one bird survive.
Let us assume that it would be possible for an enemy to create an aerosol of the causative agent of epidemic typhus ( Rickettsia prowazwki ) over City A and that a large number of cases of typhus fever resulted therefrom.
Let p be the minimal polynomial for T, Af, where the Af, are distinct irreducible monic polynomials over F and the Af are positive integers.
Let V be a finite-dimensional vector space over an algebraically closed field F, e.g., the field of complex numbers.
Let N be a positive integer and let V be the space of all N times continuously differentiable functions F on the real line which satisfy the differential equation Af where Af are some fixed constants.
Let Q be a nonsingular quadric surface bearing reguli Af and Af, and let **zg be a Af curve of order K on Q.
Let us take a set of circumstances in which I happen to be interested on the legislative side and in which I think every one of us might naturally make such a statement.
Let the state of the stream leaving stage R be denoted by a vector Af and the operating variables of stage R by Af.
Let it be granted then that the theological differences in this area between Protestants and Roman Catholics appear to be irreconcilable.
Let us therefore put first things first, and make sure of preserving the human race at whatever the temporary price may be ''.
Let and measurable
Let ( X, Σ ) and ( Y, Τ ) be measurable spaces, meaning that X and Y are sets equipped with respective sigma algebras Σ and Τ.
Let be a probability space with a filtration, for some ( totally ordered ) index set ; and let be a measurable space.
Let ( X, Σ, μ ) be a measure space, and let f be an extended real-valued measurable function defined on X.
A multiplication operator is defined as follows: Let be a countably additive measure space and f a real-valued measurable function on X.
Let U be a measurable subset of R < sup > n </ sup > and φ: U → R < sup > n </ sup > an injective function, and suppose for every x in U there exists < span > φ '</ span >( x ) in R < sup > n, n </ sup > such that φ ( y ) = φ ( x ) + < span > φ '</ span >( x ) ( y − x ) + o (|| y − x ||) as y → x.
Let X be a measurable space equipped with a measure m, and let a subset of X be negligible if it is m-null.
Let be a probability space with a filtration, for some ( totally ordered ) index set ; and let be a measurable space.
In the definition of conditional expectation that we provided above, the fact that Y is a real random variable is irrelevant: Let U be a measurable space, that is, a set equipped with a σ-algebra of subsets.
Let T be a linear mapping from the space of μ < sub > 1 </ sub >- integrable functions into the space of μ < sub > 2 </ sub >- measurable functions, and for 1 ≤ p, q ≤ ∞, define to be the operator norm of a continuous extension of T to
Let δ < sub > x </ sub > denote the Dirac measure centred on some fixed point x in some measurable space ( X, Σ ).
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