A Little Aspect of Real Analysis, Topology and Probability

The body of the paper is divided into three parts: Part one: include definitions and examples of the metric, norm and topology along with some important terms such as metrics l1,l2,l∞ and vector norm |x|p which also known as the Lp space. In case of the topology space this concept adjusted to be a unit ball that is distance one from a point unit circle. Part two: demonstrate the measure and probability which is one of the main topics in this work. This section serves as an introduction for the remaining part. It explains the construction of the σ-algebra and the Borel set, and the usefulness of the Borel set in the probability theory. Part Three: to understand the measure, one has to understand finitely additive measure and countable additivity of subsets, besides knowing the definition of ring, σ-ring and their measure extension. One also has to differentiate between the terms premeasure, outer measure and measure from their domains and additivity conditions, which is clarified in the form of table. The advantage of measurable function and the induced measure is explained, as both share to define the probability as a measure, the probability that has values in the Borel set. As a summary of this section, a Borel σ-algebra shows a special role in real-life probability because numerical data, real numbers, is gathered whenever a random experiment is performed. In this work a simple technique that is supported by pictorial presentation mostly is used, and for easiness most proofs are replaced by examples. I have freely borrowed a lot of material from various sources, and collected them in the manner that makes this thesis equipped with a little aspect of the real analysis, topology and probability theory