From this and by induction it is easy to show that the union of an finite collection of Lebesgue measurable sets is also Lebesgue measurable. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space R . They are closed under operations that one would expect for measurable sets; that is, the complement of a measurable set is a measurable set and the countable union of measurable sets is a measurable set.

Non-empty collections of sets with these properties are called σ-algebras. All open sets G [a;b] and all closed sets F [a;b] are measurable sets and m(G) = jGj, m(F) = jFj. to a set Sis measurable no matter what Ais. De nition 1.2. Because the set and its compliment is just one partition of the set. So certainly there exists a measure zero set that when added to itself gives a non-measurable set. $\blacksquare$ Corollary 2: The set $\mathcal M$ of Lebesgue measurable sets is an algebra. That is why we will demand that our measurable sets form in fact a ˙-algebra. In mathematics, a non-measurable set is a set which cannot be assigned a meaningful "volume". In practice any set you can describe in a finite English expression will be measurable. 1. The so called power set P(X), that is the collection of all subsets of X, is a ˙-algebra in X:It is simple to prove that the intersection of any family of ˙-algebras in Xis a ˙-algebra. Case 2 (Intervals of the form $(-\infty, a]$): Since every interval of the form $(a, \infty)$ is Lebesgue measurable, the complement, $(a, \infty)^c = (-\infty, a]$, is Lebesgue measurable. For Xbe a non-empty set. For instance, the Lebesgue measure of the interval[0, …

This leads us to the notion of algebra of sets.

Thus, a topological space \( (S, \mathscr S) \) naturally leads to a measurable space \( (S, \sigma(\mathscr S))\). 2. 3. That is why we will demand that our measurable sets form in fact a ˙-algebra.

The notion of a non-measurable set has been a source of great controversy since its introduction. Non-empty collections of sets with these properties are called σ-algebras.

2 Measure 2.1 De nition of Measurable set A set is measurable if it belongs to a sigma algebra Sof subsets of R. A sigma algebra Sis a collections of subsets of R such that 1.The empty set is in S. 2. The mathematical existence of such sets is construed to provide information about the notions of length, area and volume in formal set theory..