Examples of polar sets I would like to know some examples of the following polar sets (if they exist):


*

*a non trivial uncountable polar set in $\mathbb{R}^{2}$;

*a polar set in $\mathbb{R}^{2}$ contained in $\mathbb{R}$ with positive Lebesgue measure over $\mathbb{R}$.


I recall that a set $E\subset \mathbb{R}^{2}$ is called polar if there exists a subharmonic function $u:\mathbb{R}^{2}\to \mathbb{R}\cup \{-\infty\}$ such that $E\subset \{u=-\infty\}$.
 A: There is a construction mentioned in Ransford (Potential Theory in the Complex Plane, pg 143) which supplies an uncountable polar set in $\mathbb{C}$. It is reproduced below.
Letting $\{s_n\}_{n}$ be a sequence of positive numbers, $s_n<1$ and $C(s_1,\dots,s_n)$ be the set obtained by removing the middle of each interval in $C(s_1,\dots,s_{n-1})$ an open subinterval of length proportion $s_n$ of the interval (and so on, starting initially with the interval $[0,1]$), finally set 
$$C(s)=\cap_{n\geq 1} C(s_1,\dots,s_n).$$
When $s_n=1-(1/2)^{2^n}$, then $C(s)$ is polar. 
As mentioned in the comments, general Cantor sets will not work, and it is an open question to determine the capacity of the standard Cantor set, which is estimated $c(E) \sim 0.220949102189507$.
There is not a set which satisfies the second hypothesis. Any subset of the real line of positive Lebesgue measure has positive capacity and so is not polar (see Theorem 5.3.2 in Ransford).
A: A polar set of positive measure on a line does not exist. Every polar set has zero
capacity, and thus zero $1$-measure, and zero Hausdorff dimension. An uncountable polar set can be constructed
as a Cantor type set but not the standard one, the size of the segments of $n$-th generation has to decrease very fast (faster than exponentially).
Ref. Nevanlinna, Analytis functions, Ch. V, section 6,
or L. Carleson, Selected problems on exceptional sets, Van Nostrand Co., Inc., Princeton, 1967.
