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A nonnegative measurable function f is called Lebesgue integrable if its Lebesgue integral intfdmu is finite. An arbitrary measurable function is integrable if f^+ and f^- ...

The Lebesgue integral is defined in terms of upper and lower bounds using the Lebesgue measure of a set. It uses a Lebesgue sum S_n=sum_(i)eta_imu(E_i) where eta_i is the ...

The Lebesgue measure is an extension of the classical notions of length and area to more complicated sets. Given an open set S=sum_(k)(a_k,b_k) containing disjoint intervals, ...

Each double point assigned to an irreducible algebraic curve whose curve genus is nonnegative imposes exactly one condition.

The term "left factorial" is sometimes used to refer to the subfactorial !n, the first few values for n=1, 2, ... are 1, 3, 9, 33, 153, 873, 5913, ... (OEIS A007489). ...

Let A be an involutive algebra over the field C of complex numbers with involution xi|->xi^♯. Then A is a left Hilbert algebra if A has an inner product <·,·> satisfying: 1. ...

Given a map f:S->T between sets S and T, the map g:T->S is called a left inverse to f provided that g degreesf=id_S, that is, composing f with g from the left gives the ...

The (associated) Legendre function of the first kind P_n^m(z) is the solution to the Legendre differential equation which is regular at the origin. For m,n integers and z ...

Let E(k) and K(k) be complete elliptic integrals of the first and second kinds, with E^'(k) and K^'(k) the complementary integrals. Then ...

The Legendre transform of a sequence {c_k} is the sequence {a_k} with terms given by a_n = sum_(k=0)^(n)(n; k)(n+k; k)c_k (1) = sum_(k=0)^(n)(2k; k)(n+k; n-k)c_k, (2) where ...