if ∉, then the indicator function: (,) → is non-measurable (where is equipped with the Borel algebra as usual), since the preimage of the measurable set {} is the non-measurable set . Recall that B ℝ denotes the collection of Borel sets, which is the smallest σ-algebra generated by R.Thus, B R ⊂ M.Therefore, all open sets and closed sets are in M.In fact, we can see that M is a σ-algebra. of Mathematics, ETH Zurich, ariel.neufeld@math.ethz.ch . The Borel space associated to X is the pair (X,B), where B is the σ-algebra of Borel sets of X. Dept. Such a set exists because the Lebesgue measure is the completion of the Borel measure. If (,) is some measurable space and ⊂ is a non-measurable set, i.e. In general, the supremum of any countable family of measurable functions is also measurable.
This result was established in 1916 by Hausdorff and by Alexandroff, working independently. Such a model of a random G-expectation was rst introduced in [9], as an extension of the G-expectation of [15, 16]. Proof. Structure of Measurable Sets In these notes we discuss the structure of Lebesgue measurable subsets of R from sev-eral di erent points of view. Lecture #5: The Borel Sets of R We will now begin investigating the second of the two claims made at the end of Lecture #3, namely that there exists a σ-algebra B 1 of subsets of [0,1] on which it is possible to define a uniform probability.

We prove this duality formula for Borel-measurable (and, more generally, upper semianalytic) claims ˘and a model Pwhere the possible aluesv of the volatility are determined by a set-valued process. First, recall the following de nition.

Since sigma algebras are, by definition, closed under countable intersections, this shows that f is Σ-measurable.
Along the way, we will see several alternative character- izations of measurability which might help to make the concept seem more intuitive. In short, B⊂LB⊂L, where the containment is a proper one. Noticing that the Cantor set K has cardinality c and measure zero, we see that P(K) ⊂ M.On the other hand, obviously, we have M ⊂ P(ℝ). Our goal for today will be to define the Borel sets of R.Theactualconstructionofthe uniform probability will be deferred for several lectures. Our goal for today is to construct a Lebesgue measurable set which is not a Borel set. Proof. The main aim of descriptive set theory is to find structural properties common to all such definable sets: for instance, the Borel sets were shown to have the perfect set property (if uncountable, they have a perfect subset) and thus to comply with the continuum hypothesis (CH). We begin by discussing the measures of open sets. Let X be a topological space. To produce a set in L∖BL∖B, we'll assu…