Editing Extended real number line (section)

===Measure and integration===
{{confusing section|reason=since this is a subsection of section "Motivation", it must be understandable by readers who know nothing more than the basic definition of an integral|date=September 2024}}
In [[measure theory]], it is often useful to allow sets that have infinite [[measure (mathematics)|measure]] and integrals whose value may be infinite.

Such measures arise naturally out of calculus. For example, in assigning a measure to <math>\R</math> that agrees with the usual length of [[interval (mathematics)|intervals]], this measure must be larger than any finite real number. Also, when considering [[improper integral]]s, such as

:<math>\int_1^{\infty}\frac{dx}{x}</math>

the value "infinity" arises. Finally, it is often useful to consider the limit of a sequence of functions, such as      

:<math>f_n(x)=\begin{cases}2n(1-nx),&\mbox{if }0\leq x\leq\frac{1}{n}\\0,&\mbox{if }\frac{1}{n}<x\leq1\end{cases} </math>.

Without allowing functions to take on infinite values, such essential results as the [[monotone convergence theorem]] and the [[dominated convergence theorem]] would not make sense.