Real Analysis

نویسنده

  • Nikolai Chernov
چکیده

From now on we will work with sets A ⊂ X within a fixed rectangle X ⊂ R. Elementary sets in X make an algebra. Definition 1.8. The outer measure of a setA ⊂ X is μ∗(A) = inf ∑∞ i=1Area(Ri) where the infimum is taken over all countable covers of A by rectangles, A ⊂ ∪i=1Ri. Definition 1.9. The inner measure of a set A ⊂ X is μ∗(A) = Area(X)− μ∗(Ac) where A = X \ A is the complement of A. Lemma. For any elementary set B we have μ∗(B) = Area(B). Proof: BWOC, suppose μ∗(B) < Area(B). If B is a closed set covered by open rectangles, then due to compactness there is a finite subcover. If B is not closed, shrink it a bit to make it closed. If the rectangles are not open, expand them a bit to make them open.

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Some properties of fuzzy real numbers

In the mathematical analysis, there are some theorems and definitions that established for both real and fuzzy numbers. In this study, we try to prove  Bernoulli's inequality in fuzzy real numbers with some of its applications. Also, we prove two other theorems in fuzzy real numbers which are proved before, for real numbers.

متن کامل

Multi-period network Data Envelopment Analysis to measure the efficiency of a real business

Measuring the efficiency of real businesses is not a simple task, because a real business may involve several processes and sub-processes, forming a very complicated dynamic network of interactions. In this paper, a customized dynamic network data envelopment analysis (NDEA) model is proposed to measure the efficiency of the sub-processes in a real business. The proposed dynamic NDEA model is f...

متن کامل

Analysis of Herpes Simplex Virus in Suspected Encephalitis, Keratitis and Dermal Infections Using Real- Time PCR

Background & Objectives: Herpes viruses can cause diseases in the clinical range. The virus can cause infection in various body parts, especially eyes and nervous system. The aim of this study was at evaluating  the Real-Time TaqMan probe PCR in diagnosing and monitoring of the patients with suspected HSV infections.Materials & Methods: More than a thousand patients with suspected HSV infection...

متن کامل

Design and Test of the Real-time Text mining dashboard for Twitter

One of today's major research trends in the field of information systems is the discovery of implicit knowledge hidden in dataset that is currently being produced at high speed, large volumes and with a wide variety of formats. Data with such features is called big data. Extracting, processing, and visualizing the huge amount of data, today has become one of the concerns of data science scholar...

متن کامل

Combination of real options and game-theoretic approach in investment analysis

Investments in technology create a large amount of capital investments by major companies. Assessing such investment projects is identified as critical to the efficient assignment of resources. Viewing investment projects as real options, this paper expands a method for assessing technology investment decisions in the linkage existence of uncertainty and competition. It combines the game-theore...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2012