Suppose \(A\subseteq\mathbb{R}^{2}\) is Borel and \(B\) is a rectangle of \(\mathbb{R}^2\). In addition, suppose the Lebesgue measure on the Borel \(\sigma\)-algebra is \(\lambda(\cdot)\):
Question: How do we define an explicit \(A\), such that:
1. \(\lambda(A\cap B)>0\) for all \(B\)
2. \(\lambda(A\cap B)\neq\lambda(B)\) for all \(B\)?
For a potential answer, see this reddit post [1]. (It seems the answer is correct; however, I wonder if there's a simpler version that is less annoying to prove.)
Moreover, we meaningfully average \(A\) with the following approach:
Approach: We want an unique, satisfying extension of the expected value of \(A\), w.r.t the Hausdorff measure in its dimension, on bounded sets to \(A\), which takes finite values only
Question 2: How do we define "satisfying" in this approach?
(Optional: See section 3.2, & 6 of this paper [2].)
#UnboundedSets #Sets #LebesgueMeasure #MeasureTheory #Measure #ExpectedValue #Expectancy #Mean #Integration #HausdorffMeasure #HausdorffDimension