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Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.

6 votes
2 answers
1k views

Hausdorff dimension of a Cantor-like set

Suppose $K$ is a subset of $[0,1]$ with the following property: for almost $x,y \in K$, we have $$\frac{x+y}{2} \not\in K.$$ (Here, "almost in $K$" means "in $K$ except for a countable subset"). Su …
2 votes
0 answers
493 views

Characterization of weak Lebesgue spaces [closed]

I would be interested to know whether the following is true: Let $\Omega$ be a bounded open set in $\mathbf{R}^n$. Let $g$ be a nonnegative function $g : \Omega \to \mathbf{R}$. If there is a constan …