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Simple function Let $(X,\sum)$ be a measurable space, A simple function on X is a function of the form $\phi=\sum^n_{j=1}c_{j}X_Ej$, where for each $j-1,2,...,n$ an…

Measure A Measure on $(X,\sum)$ is a function $\mu:\sum\rightarrow[0,\infty]$ such that the following conditions are satisfied:

The extended real number system the extended real number system is obtained from the real number system $\mathbb{R}$ by adding two elements $+\infty$ and $-\infty$ …

Algebra $\mathbb{A}$ This is the collection $\mathcal{A}$ of subsets of X where X is a nonempty set, satisfying the following conditions. (i)$\phi\epsilon\mathcal…

The Borel Sets The Borel sets are simply defined as the elements of the borel $\sigma-algebra$. In topological space, the borel sets are sets in a topological space…

The Borel-Cantelli Lemma asserts that let (X,Σ,μ) be a measure Space, if (An) is a sequence of measurable sets such that