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Rogosinski W.W. — Volume and integral
Rogosinski W.W. — Volume and integral



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Название: Volume and integral

Автор: Rogosinski W.W.

Аннотация:

This introduction to the theory of the Lebesgue Integral is primarily intended for third-year Honours students. A consistently geometrical approach has been chosen in order to show the simple underlying ideas. First, the volume of an n-dimensional set is defined as it s Lebesgue Measure. The integral of a non-negative function of n real variables is then the volume of its ordinate set; the extension to general functions follows easily. The old notion of the Riemann Integral is similarly developed; its relation to the Lebesgue Integral and the striking advantages of the latter are clearly set out.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1952

Количество страниц: 164

Добавлена в каталог: 10.02.2013

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Предметный указатель
Absolute continuity      6.4. 6.6
algebraic      1.10
Almost everywhere      5.2
Arzela’s test      5.8
Axiom of Choice      1.14 3.9
Borel’s covering theorem      1.16
Bounded convergence      5.3 5.5
Bounded set      1.4
Bounded variation      6.4
Cantor’s set      1.15 6.6
Cauchy, integral      4.1
Characteristic function      4.3
Closed set      1.13
Closure      1.13
Co-ordinates      1.2
Complement      1.5
Congruent      2.2
Connected      1.17
Constant of integration      6.1
Content (Peano-Jordan)      2.2 2.7
Content (Peano-Jordan) of elementary sets      2.8
Content (Peano-Jordan), infinite      2.7
continuous      1.17
Convergence (of sets, points)      1.8
cube      1.3
Curve      1.17
Darboux, integrals      4.2
Darboux, integrals, theorem      4.4
Deficiencies (content)      2.9
Dense, in itself, everywhere      1.15
Dense, nowhere      1.15
Derivative of set      1.14
Derived numbers      6.4
Diameter      1.14
Difference      1.5
DIMENSION      1.2
Distance of sets      1.14
Distance, of points      1.2
Domain      1.17
Dominated convergence      5.3 5.5
EDGE      1.3
Egoroff’s theorem      5.4
Elementary sets      2.2
Empty set      1.3
Enumerable      1.9
Equivalent      5.2
Euclidean space      1.1
Eventually      1.8
exclusive      1.6
Extended function      4.2
Exterior      1.12
Factor      1.6
Fatou’s lemma, integrals      5.3
Fatou’s lemma, sets      3.8
Finite set      1.3
Frontier      1.12
Fubini’s Theorem      5.7
Function      1.17
Graph      2.8
Image      1.17
Infinite set      1.3
Inner content      2.6
Inner measure      3.5 3.6
Integrable      4.1 4.3 5.2 5.5
Integral (Lebesgue)      5.1 5.2 5.8
Integral (Riemann)      4.3 5.8
Integral, definite, absolute      4.1
Integral, improper (Cauchy)      4.1
Integral, indefinite      6.1 6.6
Integration by parts      6.7
Interior      1.12
interval      1.3
Interval sets      3.2
Interval sums      2.3
Interval sums, general      2.8
Interval, general      2.8
Isolated point      1.12
Jordan curve      1.17
Lattice points      1.10
Lebesgue integral      6.6
Lebesgue, division      5.6
Lebesgue, integrable      5.2 5.5
Lebesgue, integral      5.1 5.2 5.8
Lebesgue, measure      3.7
Lebesgue, sums      5.6
Lebesgue, test      5.3
Limit (of sequences)      1.8 5.3
Limit of sets      1.8
Limit, inferior, superior (of sets)      1.8 3.8
Limiting point      1.14
Limiting properties (of L-integral)      5.3 5.5
Limiting properties (of ^-measure)      3.8
Limiting sets      1.8 3.8
Mean value theorem (integrals), 1st      5.5
Mean value theorem (integrals), 2nd      6.3
Measurable      3.7
Measurable functions      5.4 5.5
Measure (Lebesgue)      3.7
Neighbourhood      1.12
Non-enumerable      1.11
Null set      1.3
Open set      1.13
Ordinate set      4.1
Oscillation      4.4
Outer content      2.4 2.5
Outer measure      3.4
Overlap      1.6
P.p. (presque partout)      5.2
Perfect      1.15
Point      1.2
Polygon      2.8
Polyhedron      2.8
Primary set      2.2
Primitive function      6.1
PRODUCT      1.6
Projection      5.7
Rational set      1.10 2.9 5
Region      1.17
Riemann integral      4.5 6.6
Riemann, criterion      4.4
Riemann, division      4.4
Riemann, integrable      4.3
Riemann, integral      4.3
Riemann, lower, upper integral      4.2
Riemann, sums      4.4
Section      5.7
separate      2.3
Separated      2.4
SEQUENCE      1.8
Set      1.3
Sphere      1.3
Subset      1.4
subspace      1.4
Substitution rule      6.7
SUM      1.7
surface      1.17
Term      1.7
Transformation      1.17
Triangle relation      1.2
Uniform continuity      1.17
Uniform convergence      4.5
union      1.7
Variation      6.4
Vitali, covering theorem      3.3
Vitali, set      3.9 5.2
Volume, interval      1.3
Volume, interval set      3.3
Volume, interval sum      2.3
Volume, postulates      2.2 3.1
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