A Primer of Lebesgue IntegrationAcademic Press, 2002 - 164 Seiten The Lebesgue integral is now standard for both applications and advanced mathematics. This books starts with a review of the familiar calculus integral and then constructs the Lebesgue integral from the ground up using the same ideas. A Primer of Lebesgue Integration has been used successfully both in the classroom and for individual study. Bear presents a clear and simple introduction for those intent on further study in higher mathematics. Additionally, this book serves as a refresher providing new insight for those in the field. The author writes with an engaging, commonsense style that appeals to readers at all levels. |
Inhalt
| 93 | |
General Measures | 107 |
Integration for General Measures | 117 |
The RadonNikodym Theorem | 127 |
Product Measures | 135 |
The Space L² | 149 |
Index | 161 |
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Häufige Begriffe und Wortgruppen
A₁ admissible partition assume B₁ bounded function Cantor set Chapter countable family countable union countably additive defined definition directed set disjoint measurable sets dyadic squares dµ(x E₁ f is bounded f is continuous f is integrable f is measurable F₁ Fatou's Lemma finite measure set finite number finite or countable fn(x Fubini Theorem function f ƒ and g Hence Hint integrable function intersect Lebesgue integrable Let f lim inf lim R(f linear lower sums measurable functions measurable sets measurable subsets measure zero Monotone Convergence Theorem negative set null set o-algebra o-finite outer measure P₁ plane measure pointwise positive measure Problem 11 Riemann integrable Riemann sums sequence of measurable set of finite set of measure Show signed measure simple functions Ss f subadditive surable union of rectangles upper sums
Beliebte Passagen
Seite 152 - ... (i) d(x, y) ^ 0 and d(x, y} = 0 if and only if x...
Seite 45 - If f is Riemann integrable on [a, b], then f is Lebesgue integrable on [a, b] and the integrals are the same.
Seite 35 - Show that for each e > 0 there is an open set U and a closed set F such that F c £ CI/ and m(E) -e < m(F) < m(U) < m(E} + e.
Seite 53 - The inequality f(x) + g(x) > a is equivalent to f(x) > a — g(x), which holds if and only if there is a rational number r such that f(x) > r and r > a — g(x).
Seite 3 - J and that this inequality is a consequence of the fact that every lower sum is less than or equal to every upper sum: If Ж and 31 are any two nets on [а, Ь], (8) L(SfR) а i/(3l).
Seite 10 - Ja property of addition: if a is close to A and b is close to B, then a + b is close to A + B.
Seite 60 - If f is bounded on [a, b], then f is Riemann integrable on [a . b] if and only if f is continuous almost everywhere, ie, the set where f is discontinuous has measure zero.
Seite 4 - Proposition 3. f is integrable on [a, b] if and only if for each £>0 there is a partition P of [a,b] such that U(f,P) — L(f,P)<£.
Seite 5 - If f is bounded on [a, b] and continuous except at a finite number of points, then f is integrable on [a, b].
Verweise auf dieses Buch
A Radical Approach to Lebesgue's Theory of Integration David M. Bressoud Eingeschränkte Leseprobe - 2008 |