Length of an Interval II lebesgue measure II Real Analysis

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More precisely, the length of an interval is the absolute value of the difference of the two endpoints. E.g: For the interval [3, 10], the length is |10 - 3| = |3 - 10| = 7.

Here we discuss about
What is open and bounded intervals

length of open and bounded interval

Which comes in the outer measure

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The length 1(I) of an interval I is.defined to be the difference of the endpoints of I
if I is bounded, and oo if I is unbounded. Length is an example of a set function, that is, a
function that associates an extended real number to each set in a collection of sets. In the
case of length, the domain is the collection of all intervals. In this chapter we extend the set
function length to a large collection of sets of real numbers. For instance, the "length" of an
open set will be the sum of the lengths of the countable number of open intervals of which
it is composed. However, the collection of sets consisting of intervals and open sets is still
too limited for our purposes. We construct a collection of sets called Lebesgue measurable
sets, and a set function of this collection called Lebesgue measure which is denoted by m.