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].)

[1]: https://www.reddit.com/r/mathematics/comments/1eedqbx/is_there_a_set_with_positive_lebesgue_measure_in/

[2]: https://www.researchgate.net/publication/382994255_Finding_a_Published_Research_Paper_Which_Meaningfully_Averages_The_Most_Pathalogical_Sets

#UnboundedSets #Sets #LebesgueMeasure #MeasureTheory #Measure #ExpectedValue #Expectancy #Mean #Integration #HausdorffMeasure #HausdorffDimension

HÖLDER'S INEQUALITY:
If \(\mathcal{S}\) is a measurable subset of \(\mathbf{R}^n\) with the Lebesgue measure, and \(\eta\) and \(\xi\) are measurable real- or complex-valued functions on \(\mathcal{S}\), then

\[\displaystyle\left(\int_\mathcal{S}|\eta(x)\xi(x)|\ \mathrm{d}x\right)^{ab}\leq\left(\int_\mathcal{S}|\eta(x)|^a\ \mathrm{d}x\right)^b\left(\int_\mathcal{S}|\xi(x)|^b\ \mathrm{d}x\right)^a\]
where, \((a,b)\in(1,\infty)^2\) with \(1/a+1/b=1\).
#HolderInequality #LebesgueMeasure #Integrals