| Abstract | U ovom diplomskom radu izgrađivali smo Jordanovu mjeru i njezina svojstva. Na početku, u Poglavlju 1, aksiomatski smo zadali ravninu te definirali dužinu, konveksan skup, konveksnu ljusku i poluravnine. Dokazali smo njihova svojstva koja ćemo koristiti u daljnjim poglavljima. U Poglavlju 2 uveli smo pojam trokuta kao konveksne ljuske skupa koji sadrži tri nekolinearne točke, te pojam poligona kao uniju konačno mnogo trokuta. Definirali smo rub trokuta, unutrašnjost trokuta i promatrali što vrijedi za međusobne odnose dvaju trokuta, pravca i trokuta i njihovih unutrašnjosti. U Poglavlju 3 razradili smo pojmove kompleksa dužina, segmentacije skupa, triangulacije skupa te kompleksa i polukompleksa trokuta. Zatim smo u Poglavlju 4 uveli funkcije koje računaju površinu trokuta i poligona. U završnom dijelu, Poglavlju 5, definirali smo broj koji nazivamo Jordanova mjera skupa s obzirom na poligonsku površinu. U ovom smo poglavlju proučavali i svojstva Jordanove mjere. |
| Abstract (english) | In this thesis we constructed the Jordan measure and its properties. At the beginning, in Chapter 1, we have axiomatically constructed a plane and defined the segment, convex set, convex hull and half-plane. We have proved their properties which we will be using in further chapters. In Chapter 2, we have introduced the concept of a triangle as the convex hull of a set which contains three non collinear points, and the concept of a polygon as a union of finite number of triangles. We have defined a border of the triangle, its interior, and observed what is valid for relations between line and triangle, two triangles and their interiors. In Chapter 3, we have developed concepts of complex of segments, segmentation of a set, triangulation of a set and a complex of triangles. Then in Chapter 4, we have introduced functions which calculate area of the triangle and polygon. In the final chapter, Chapter 5, we have defined a number which is called the Jordan measure of a set, considering the polygon area. In this final chapter, we have also studied the properties of the Jordan measure. |
