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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 ...
The Lebesgue integral describes better how and when it is possible to take limits under the integral sign (via the monotone convergence theorem and dominated convergence theorem). While the Riemann integral considers the area under a curve as made out of vertical rectangles, the Lebesgue definition considers horizontal slabs that are not ...
The substitution is described in most integral calculus textbooks since the late 19th century, usually without any special name. [5] It is known in Russia as the universal trigonometric substitution , [ 6 ] and also known by variant names such as half-tangent substitution or half-angle substitution .
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.
Dominated convergence theorem (Lebesgue integration) Egorov's theorem (measure theory) Fatou–Lebesgue theorem (real analysis) Fubini's theorem (integration) Hahn decomposition theorem (measure theory) Hahn–Kolmogorov theorem (measure theory) Ham sandwich theorem ; Hobby–Rice theorem (mathematical analysis) Kōmura's theorem (measure theory)
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.