Question

a set E is mesureable if its outr measure is equal to (a) inner measure (b)outr measurr​

Answer 1

Answer:

e mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer measures was first introduced by Constantin Carathéodory to provide an abstract basis for the theory of measurable sets and countably additive measures.[1][2] Carathéodory's work on outer measures found many applications in measure-theoretic set theory (outer measures are for example used in the proof of the fundamental Carathéodory's extension theorem), and was used in an essential way by Hausdorff to define a dimension-like metric invariant now called Hausdorff dimension. Outer measures are commonly used in the field of geometric measure theory.

Measures are generalizations of length, area and volume, but are useful for much more abstract and irregular sets than intervals in R or balls in R3. One might expect to define a generalized measuring function φ on R that fulfils the following requirements:

Any interval of reals [a, b] has measure b − a

The measuring function φ is a non-negative extended real-valued function defined for all subsets of R.

Translation invariance: For any set A and any real x, the sets A and A+x have the same measure (where {\displaystyle A+x=\{a+x:a\in A\}}A+x = \{ a+x: a\in A\})

Countable additivity: for any sequence (Aj) of pairwise disjoint subsets of R

{\displaystyle \varphi \left(\bigcup _{i=1}^{\infty }A_{i}\right)=\sum _{i=1}^{\infty }\varphi (A_{i}).} \varphi\left(\bigcup_{i=1}^\infty A_i\right) = \sum_{i=1}^\infty \varphi(A_i).