Is there an open connected set in $\mathbb{R^2}$ of finite Lebesgue measure, which has a non-empty intersection with any straight line? Is there an open connected set in $\mathbb{R^2}$ of finite Lebesgue measure, which has a non-empty intersection with any straight line ?
My Try :
The set of tuples $\Gamma = \{ (q_k,p_k) \}$ is a countable and dense subset of $\mathbb{R^2}$. Let $\{ r_1,r_2,... \}$ be an enumeration of $\Gamma$. For each $k \in \mathbb{N}$ Consider the open square $S_k$ having $r_k$ at its central point, with sides having length of $\sqrt{\frac{\epsilon}{2^k}}$. Hence , $m(S_k)=\frac{\epsilon}{2^k}$. Now define $O=\bigcup^{\infty}_{k=1} S_k$, $$m(O) \leq \sum^{\infty}_{k=1} m(S_k) = \sum^{\infty}_{k=1} \frac{\epsilon}{2^k}=\epsilon$$
$O$ has a nonempty intersection with any line in $\mathbb{R^2}$ according to density of $\mathbb{Q^2}$ in $\mathbb{R^2}$ .
Seemingly, I have come up with an open set, dense in $\mathbb{R^2}$, and having as small measure as I want. Isn't it too much ?! That's why I decided to ask it here\, because I was asked to find an open set of "finite" measure.
 A: Consider the set
$$A = \left\{(x, y) : |y| < \frac{1}{1 + x^2}\right\}$$
This has finite Lebesgue measure; its measure is twice the integral (over $\mathbb{R}$) of $1/(1 + x^2)$. Add to it a rotation of $A$ by an angle of $90^{\circ}$ around the origin, so that we have two double-sombrero shape regions. This new set satisfies the conditions posed.
A: A straight line has to intersect either the x-axis or the y-axis, so just take an open connected set covering the x and y axes with finite measure. For example for the x-axis, just take open boxes $B_n$ of length and height $\frac{1}{n}$, and make sure they each intersect a little bit, and similarly for the y-axis.
A: Try this.  Let $\epsilon > 0$. Now let $U$ be the set of all $(x,y)$ so $|y| < \epsilon e^{-|x|}$.  Let $V$ be its reflection through the line $y = x$.  Put $G = U\cup V$. This can be made to have arbitrarily small measure and meets any line in the plane.
A: Perhaps more interesting is that you can have a connected open set of finite (and arbitrarily small) Lebesgue measure that intersects every straight line segment (of nonzero length).  
Hint: every such segment intersects at least one of the lines $x = r$ and $y=r$ for rational $r$.
