query: outer measure of an irrational set
positive_passages.docid: 19289
positive_passages.text: complement of a set and its measure
negative_passages.docid: 134310
negative_passages.text: Minkowski sum of a positive Lebesgue measure set and $\mathbb{Q}$.
negative_passages.docid: 81150
negative_passages.text: Positive integer multiples of an irrational mod 1 are dense
negative_passages.docid: 133500
negative_passages.text: For an outer measure $m^*$, does $m^*(E\cup A)+m^*(E\cap A) = m^*(E)+m^*(A)$ always hold?
negative_passages.docid: 130981
negative_passages.text: Co-countable measure on uncountable set
negative_passages.docid: 13550
negative_passages.text: For an irrational number $\alpha$, prove that the set $\{a+b\alpha: a,b\in \mathbb{Z}\}$ is dense in $\mathbb R$
negative_passages.docid: 28468
negative_passages.text: Haar measure on $SO(n)$
negative_passages.docid: 30523
negative_passages.text: A binary irrational with bits defined by primes
negative_passages.docid: 83825
negative_passages.text: $f^{-1}([-\infty,r))$ measurable for all rational numbers $r$
negative_passages.docid: 65595
negative_passages.text: Let $\lambda(A)$ be the Lebesgue measure of $A$. There exists an interval $I$ such that $\lambda(E \cap I) / \lambda(I) < 1-\epsilon$
negative_passages.docid: 72050
negative_passages.text: Is Lebesgue outer measure continuous from above for sets with finite Lebesgue outer measure
negative_passages.docid: 23752
negative_passages.text: Proving Irrationality
negative_passages.docid: 18360
negative_passages.text: Proof that the irrational numbers are uncountable
negative_passages.docid: 70409
negative_passages.text: Rational Equivalence Relation and Measure
negative_passages.docid: 95294
negative_passages.text: Dense subset of $C[0,1]$ which preserves rationals
negative_passages.docid: 136795
negative_passages.text: A non-measurable set of given outer measure
negative_passages.docid: 113227
negative_passages.text: Measure spaces are proper subsets