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Funkcijski prostori, odvod in integral : magistrsko delo
ID Vidmar, Gloria (Author), ID Slapar, Marko (Mentor) More about this mentor... This link opens in a new window

URLURL - Presentation file, Visit http://pefprints.pef.uni-lj.si/4396/ This link opens in a new window

Abstract
V magistrskem delu bo najprej predstavljena Lebesgueova mera. Vpeljali jo bomo preko zunanje mere z zahtevanjem pogoja števne aditivnosti. Predstavljene bodo tudi lastnosti merljivih množic in funkcij, pri katerih bo poudarek na stopničastih in enostavnih funkcijah, s katerimi lahko definiramo Riemannov in Lebesgueov integral. Sledilo bo nekaj lastnosti Riemannovega integrala in vpeljava Lebesgueovega integrala, pri čemer se bomo sklicevali na prej obravnavano mero in merljive funkcije. V glavnem delu se bomo posvetili nekaterim razredom zveznih funkcij, predvsem skoraj povsod odvedljivim funkcijam, funkcijam z omejeno variacijo in absolutno zveznim funkcijam. Omenjene bodo tudi nikjer odvedljive, monotone in Lipschitzove funkcije. Predstavila bom, kaj lahko trdimo za posamezen razred, tudi kar se tiče Riemannove in Lebesgueove integrabilnosti. Zadnji del bo namenjen obravnavi Lp prostorov, kjer bomo pokazali, da so ti prostori normirani, polni, in da je razred enostavnih funkcij gost podprostor prostora Lp.

Language:Slovenian
Keywords:Lebesgueova mera, Lebesgueov integral, funkcije z omejeno variacijo, absolutno zvezne funkcije, osnovni izrek integralskega računa, Lp prostori
Work type:Master's thesis/paper
Typology:2.09 - Master's Thesis
Organization:PEF - Faculty of Education
Publisher:[G. Vidmar]
Year:2017
Number of pages:46 str.
PID:20.500.12556/RUL-91132 This link opens in a new window
UDC:51(043.2)
COBISS.SI-ID:11488585 This link opens in a new window
Publication date in RUL:30.08.2017
Views:1562
Downloads:313
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Secondary language

Language:English
Title:Function spaces, derivative and integral
Abstract:
In this master's thesis we first presents the Lebesgue measure, which is introduced through an outer measure satisfying the condition of countable additivity. We also discuss the properties of Lebesgue measurable sets, focusing on step functions and simple functions that can be used to define the Riemann and Lebesgue integrals. Next we discuss the properties of the Riemann integral and introduce the Lebesgue integral, while referring to the properties of the measure and measurable functions. The main part of the master’s thesis revolves around some classes of continuous functions, especially almost everywhere differentiable functions, functions with bounded variation and absolutely continuous functions. It also presents nowhere differentiable functions, monotonic functions and the Lipschitz functions. Furthermore, we discuss the properties of these individual classes of functions, including their Riemann and Lebesgue integrability. The final part focuses on Lp spaces where we show that they are normed and complete vector spaces and that the class of simple functions is, in fact, a dense subspace of an Lp space.

Keywords:mathematics, matematika

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