Funkcijski prostori, odvod in integral

magistrsko delo

Gloria Vidmar (Author), Marko Slapar (Mentor)

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.

Keywords

Lebesgueova mera;Lebesgueov integral;funkcije z omejeno variacijo;absolutno zvezne funkcije;osnovni izrek integralskega računa;Lp prostori;

Data

Language:	Slovenian
Year of publishing:	2017
Typology:	2.09 - Master's Thesis
Organization:	UL PEF - Faculty of Education
Publisher:	[G. Vidmar]
UDC:	51(043.2)
COBISS:	11488585
Views:	689
Downloads:	200
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Function spaces, derivative and integral
Secondary 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.
Secondary keywords:	mathematics;matematika;
File type:	application/pdf
Type (COBISS):	Master's thesis/paper
Thesis comment:	Univ. v Ljubljani, Pedagoška fak., Poučevanje, Predmetno poučevanje
Pages:	46 str.
ID:	9593026