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master's thesis
Radon - Nikodymova derivacija
Split: University of Split, Faculty of Science, 2023. urn:nbn:hr:166:533695
Mikulić, Lucia
University of Split
Faculty of Science
Department of Mathematics

Institutional repository: Repository of Faculty of Science

Cite this document

Mikulić, L. (2023). Radon - Nikodymova derivacija (Master's thesis). Split: University of Split, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:166:533695

Mikulić, Lucia. "Radon - Nikodymova derivacija." Master's thesis, University of Split, Faculty of Science, 2023. https://urn.nsk.hr/urn:nbn:hr:166:533695

Mikulić, Lucia. "Radon - Nikodymova derivacija." Master's thesis, University of Split, Faculty of Science, 2023. https://urn.nsk.hr/urn:nbn:hr:166:533695

Mikulić, L. (2023). 'Radon - Nikodymova derivacija', Master's thesis, University of Split, Faculty of Science, accessed 22 July 2024, https://urn.nsk.hr/urn:nbn:hr:166:533695

Mikulić L. Radon - Nikodymova derivacija [Master's thesis]. Split: University of Split, Faculty of Science; 2023 [cited 2024 July 22] Available at: https://urn.nsk.hr/urn:nbn:hr:166:533695

L. Mikulić, "Radon - Nikodymova derivacija", Master's thesis, University of Split, Faculty of Science, Split, 2023. Available at: https://urn.nsk.hr/urn:nbn:hr:166:533695

Metadata
TitleRadon - Nikodymova derivacija
Title (english)Radon - Nikodym derivative
AuthorLucia Mikulić
MentorVesna Gotovac Đogaš (mentor)
Committee memberVesna Gotovac Đogaš (predsjednik povjerenstva)
Committee memberAna Laštre (član povjerenstva)
Committee memberIvan Jelić (član povjerenstva)
GranterUniversity of Split
Faculty of Science
(Department of Mathematics)
Split
Defense date and country2023-09-29, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
Radon-Nikodymova teorija derivacija se smatra temeljem moderne teorije uvjetne vjerojatnosti. Na početku rada javlja se pitanje odnosa operacija integracije i diferencijacije. Definiranjem, između ostalog, disjunktnih nosača i apsolutne neprekidnosti dobili smo karakterizaciju. Funkcija distribucije F ima oblik \(\int_{-\infty}^{x} F'(t)dt\) ako i samo ako je apsolutno neprekidna. U drugom dijelu rada cilj je pokazati da ako su ν i µ σ-konačne mjere na (Ω, F, µ), onda ν ima gustoću f u odnosu na µ. Sada vidimo da smo tu tvrdnju prethodno dokazali za (\(\mathbb{R}^{1},\mathbb{B}(\mathbb{R}^{1}), λ\)) i za uzete mjere µ i λ. Problem je poznat i kao Radon-Nikodymov teorem. Za ν koja je apsolutno neprekidna u odnosu na µ, postoji gustoća f koja je nenegativna. Naziv tražene gustoće je upravo Radon-Nikodymova derivacija.
Abstract (english)
Radon-Nikodym’s theory of derivatives underlies the modern theory of conditional probability. At the beginning, the question of the relationship between integration and differentiation operations arises. By defining, among other things, disjoint supports and absolute continuity, we obtained a characterization. The distribution function F has the form \(\int_{-\infty}^{x} F'(t)dt\) if and only if it is absolutely continuous. In the second part, the goal is to show that if ν and µ are σ-final measures on (Ω, F, µ), then ν has density f compared to µ. Now we see that this statement was previously proved for (\(\mathbb{R}^{1},\mathbb{B}(\mathbb{R}^{1}), λ\)) and for the taken measures µ and λ. The problem is also known as the Radon-Nikodym theorem. For ν which is absolutely continuous with respect to µ, there exists a density f which is non-negative. The name of the desired density is precisely the Radon-Nikodym derivative.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:166:533695
Study programmeTitle: Mathematics; specializations in: major in pure mathematics, major in computing, major in teaching, major in statistics and computing
Course: major in statistics and computing
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence Attribution 4.0 International (CC BY 4.0) icon
Created on2023-10-03 11:45:30