This is a reference request. 


The phrase in the title is, if I remember correctly, how Eli Stein described the following set (the definition may be faulty, but I think it is right): 



> 
> There exists a set $S$, which is a subset of the unit square $[0,1]^2\subset\mathbb{R}^2$ with **full Lebesgue measure**, with the property that for any point $p\in S$ there exists a line $\ell \subset \mathbb{R}^2$ such that $\ell \cap S = \{ p\}$. 
> 
> 
> 


I vaguely remember the theorem being attributed to one of the Rieszes (not sure M. or F.; I might also be completely wrong about that). 


Can someone tell me whether my rough statement of the theorem above is correct, and better yet, can someone give a reference to the paper which proved it? (I know a paper exists; I remember digging it up in the bowels of the library when I was a beginning graduate student. I am sad to say that two moves later I can no longer find my photocopy of that article.)



"A sea-side town where every house can see the sea"
This is a [Nikodymn set](http://en.wikipedia.org/wiki/Nikodym_set). I haven't seen a citation to Nikodymn's original paper, but the history is breifly discussed (with references) in Stein's survey article [Singular Integrals: The Roles of Calderón and Zygmund](http://www.ams.org/notices/199809/stein.pdf).


Edit: Stein's Harmonic Analysis cites this to: "O. Nikodym, Sur les ensembles accessibles, Fund. Math. 1927 10:116-168"


