Luzinov teorem za kapacitet

master's thesis

• Title (english): Luzin's capacity theorem
• Author: Toni Zlatunić
• Mentor: Zoran Vondraček (mentor)
• Committee member: Zoran Vondraček (predsjednik povjerenstva)
• Committee member: Dražen Adamović (član povjerenstva)
• Committee member: Tomislav Berić (član povjerenstva)
• Committee member: Bojan Basrak (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-12-01, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Cilj ovog diplomskog rada bio je dati nužne i dovoljne uvjete da bi vrijedio Luzinov teorem za kapacitet. Luzinov teorem je važan teorem iz teorije mjere, no u ovom radu mjeru zamjenjujemo kapacitetom. Riječ je o funkciji koja ima svojstva slična mjeri, ali malo slabija. Na početku uvodimo sam pojam kapaciteta te navodimo neka njegova svojstva te primjere istih. Nakon toga dokazujemo nekoliko rezultata iz matematičke analize, metričkih prostora i teorije mjere koji su nam potrebni za dokaz glavnog rezultata. Također, uvodimo i važan pojam regularnog kapaciteta te pokazujemo u kakvoj je vezi taj pojam s neprekidnošću odozgo na nizove otvorenih skupova. U glavnom dijelu ovog rada dokazujemo da je upravo neprekidnost odozgo na nizove otvorenih skupova ključno svojstvo koje subaditivan kapacitet na kompaktnom metričkom prostoru mora posjedovati da bi vrijedio Luzinov teorem za tu klasu kapaciteta. Zapravo pokazujemo i više, odnosno da je to i nužan i dovoljan uvjet za Luzinov teorem. Na kraju ovog rada pokazujemo dvije posljedice Luzinovog teorema za kapacitet koje govore o ograničenosti izmjerivih funkcija te o aproksimaciji proizvoljne izmjerive funkcije neprekidnom funkcijom.

Abstract (english)

The aim of this master’s thesis was to give necessary and sufficient conditions for Lusin’s theorem for capacity. Lusin’s theorem is an important theorem of measure theory, but in this thesis, we replace the measure with capacity. The capacity is a function that has properties similar to the measure, but slightly weaker. In the introductory part, we define the concept of the capacity, present some of its properties and give some examples of capacities. After that, we prove several results from mathematical analysis, metric spaces and measure theory that we need for the proof of the main result. Also, we define an important concept of regular capacity and show in what relation is the regularity with the continuity from above on the sequences of open sets. In the main part of the thesis, we prove that the continuity from above on the sequences of open sets is the crucial property that subadditive capacity on the compact metric space must possess for Lusin’s theorem to hold true for this class of capacities. In fact, we show even more, that is, we show that it is a necessary and sufficient condition for Lusin’s theorem. Lastly, we show two consequences of Lusin’s theorem for capacity concerning the boundness of measurable functions and approximation of an arbitrary measurable function with a continuous function.

• Keywords: teorija mjere; matematička analiza; metrički prostori; regularni kapacitet; subaditivan kapacitet
• Keywords (english): measure theory; mathematical analysis; metric spaces; regular capacity; subadditive capacity
• Language: croatian
• URN:NBN: urn:nbn:hr:217:201247
• Study programme: Title: Mathematical Statistics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-12-17 15:30:10