Question 1: In $\mathbb{R}^2$, let $l\_1$,$l\_2$ be two parallel lines and $l\_3$ another line which is not parallel to $l\_1$. Given two measurable sets $E\_1$ and $E\_2$ in $l\_1$ and $l\_2$ respectively, both of which have positive (1-dimension) Lebesgue measure. Denote by $l\_{u,v}$ the line passing through two points $u,v$. Now, we define the third set $E\_3$ in $l\_3$ as follows.
$E\_3=\lbrace{p\in l\_3: p=l\_{u,v}\cap l\_3, u\in E\_1, v\in E\_2\rbrace}$.


Can we say that $E\_3$ contains an interval (or say "a segment") in $l\_3$?


EDIT. Joseph Van Name answered Question 1 affirmatively. The following Question 2 is a generalization of Question 1, which is not answered.


Question 2: In $\mathbb{R}^n$, let $L\_1$,$L\_2$ be two parallel (n-1)-dimensional hyperplanes and $L\_3$ another (n-1)-hyperplane which is not parallel to $L\_1$. Given two measurable sets $E\_1$ and $E\_2$ in $L\_1$ and $L\_2$ respectively, both of which have positive ((n-1)-dimension) Lebesgue measure. Denote by $l\_{u,v}$ the line passing through two points $u,v$. Now, we define the third set $E\_3$ in $L\_3$ as follows.
$E\_3=\lbrace{p\in L\_3: p=l\_{u,v}\cap L\_3, u\in E\_1, v\in E\_2\rbrace}$.


Can we say that $E\_3$ contains an $(n-1)$-ball in $L\_3$?


It seems that the result in "An elementary proof and an extension of a theorem of Steinhaus,Kuczma, Marcin E.; Kuczma, Marek, Glasnik Mat. Ser. III 6(26) (1971), 11–18." noted by Joseph Van Name does not apply to the general case $n\geq 3$ of Question 2.



Operation on measurable sets in lines, containing an interval?
We will need the following result from the paper that I mentioned in the comments "An elementary proof and an extension of a theorem of Steinhaus" by M. Kuczma and M. Kuczma that generalizes the Steinhaus theorem.


Thm:(Kuczma and Kuczma) Let $A,B$ be sets with positive inner Lebesgue measure and let $f(x,y)$ be a real-valued function of class $C^{1}$ in a region $D\supseteq A\times B$ with $f\_{x}\neq 0,f\_{y}\neq 0$ in $D$. Then the set $f(A\times B)$ contains an interval.


Now, to prove our result we shall assume without loss of generality that $\ell\_{1}=\{(x,0)|x\in\mathbb{R}\},\ell\_{2}=\{(x,1)|x\in\mathbb{R}\}$ and $\ell\_{3}=\{(0,y)|x\in\mathbb{R}\}$. Let $U=\{(x,y)\in\mathbb{R}^{2}|x\neq y\}$. Then $U$ is the union of the two regions $\{(x,y)|x<y\}$ and $\{(x,y)|x>y\}.$ Let $f:U\rightarrow\mathbb{R}$ be the function where if $(x,y)\in U$, then $f(x,y)$ is the unique real number such that the points $(x,0),(y,1),(0,f(x,y))$ all fall on the same line. Then any reasonable high school algebra student should be able to tell you that $f(x,y)=\frac{x}{x-y}$. Therefore $f$ is real-analytic on $U$. Also, any calculus 3 student should also be able to tell you that $f\_{x}(x,y)=\frac{-y}{(x-y)^{2}}$ and $f\_{y}(x,y)=\frac{x}{(x-y)^{2}}$.` Therefore the first partial derivatives of $f$ with respect to $x$ and $y$ do not vanish on the set $U$.


If $E\_{1},E\_{2}\subseteq\mathbb{R}$ are sets with positive Lebesgue measure, then there are subsets $A\_{1}\subseteq E\_{1},A\_{2}\subseteq E\_{2}$ of positive Lebesgue measure such that either $a\_{1}<a\_{2}$ for $a\_{1}\in A\_{1},a\_{2}\in A\_{2}$ or $a\_{1}>a\_{2}$ for $a\_{1}\in A\_{1},a\_{2}\in A\_{2}$. Therefore the set $A\_{1}\times A\_{2}$ is completely contained in some region in $U$, so $f(A\_{1}\times A\_{2})$ contains some interval. We therefore conclude that the set $E\_{3}$ defined in the problem must contain some interval as well.


