The following is problem 7 from chapter 1 ("Measure Theory") of Stein and Shakarchi's Real Analysis.
Consider the curve $\Gamma = \{y = f(x)\}$ in $\mathbb{R}^2$ , $0 \leq x \leq 1$. Assume that $f$ is twice continuously differentiable in $0 \leq x \leq 1$. Then show that $m(\Gamma + \Gamma) > 0$ if and only if $\Gamma + \Gamma$ contains an open set, if and only if $f$ is not linear.
I have proved that $\Gamma + \Gamma$ contains an open set if and only if $f$ is not linear, through the map $$\varphi\colon(a,b)\mapsto(a+b,f(a)+f(b)),$$
which is locally differential homeomorphic when $f$ is not linear.
And I also found that for given $0\leq t\leq 2$, the image of $(t,f(h)+f(t-h))$ is a line segment. Hence $\Gamma+\Gamma$ is path-connected.
But now I even do not know whether $\Gamma + \Gamma$ is measurable or not. And I do not know the why there is the condition "twice continuously differentiable".
Could you help me solve the problem? Thank you!
Best Answer
Recall that $\varphi (a,b)=(a+b,f(a)+f(b))$.
$[0,1]^2$ is compact and $\varphi$ is continuous, so $\Gamma+\Gamma=\varphi([0,1]^2)$ is also compact.