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The substitutions of Euler can be generalized by allowing the use of imaginary numbers. For example, in the integral +, the substitution + = + can be used. Extensions to the complex numbers allows us to use every type of Euler substitution regardless of the coefficients on the quadratic.
For Lebesgue measurable functions, the theorem can be stated in the following form: [6] Theorem — Let U be a measurable subset of R n and φ : U → R n an injective function , and suppose for every x in U there exists φ ′( x ) in R n , n such that φ ( y ) = φ ( x ) + φ′ ( x )( y − x ) + o (‖ y − x ‖) as y → x (here o is ...
A natural "Lebesgue measure" on the unit circle S 1 (here thought of as a subset of the complex plane C) may be defined using a push-forward construction and Lebesgue measure λ on the real line R. Let λ also denote the restriction of Lebesgue measure to the interval [0, 2 π ) and let f : [0, 2 π ) → S 1 be the natural bijection defined by ...
The theorem also holds if balls are replaced, in the definition of the derivative, by families of sets with diameter tending to zero satisfying the Lebesgue's regularity condition, defined above as family of sets with bounded eccentricity. This follows since the same substitution can be made in the statement of the Vitali covering lemma.
In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou. Fatou's lemma can be used to prove the Fatou–Lebesgue theorem and Lebesgue's dominated convergence theorem.
The result for Lebesgue measure turns out to be a special case of the following result, which is based on the Besicovitch covering theorem: if μ is any locally finite Borel measure on R n and f : R n → R is locally integrable with respect to μ, then (()) () = for μ-almost all points x ∈ R n.
A Lebesgue-measurable set can be "squeezed" between a containing G δ set and a contained F σ. I.e, if A is Lebesgue-measurable then there exist a G δ set G and an F σ F such that G ⊇ A ⊇ F and λ(G \ A) = λ(A \ F) = 0. Lebesgue measure is both locally finite and inner regular, and so it is a Radon measure.
In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations, which concerns finding functions that optimize the values of quantities that depend on those functions.