# Set of lower dimension has Lebesgue measure zero?

Let $$f(x):\mathbb{R}\to\mathbb{R}$$ be a continuous strictly monotonic function. For example, $$f(x)=\tanh(x)$$.

Does the set $$\{\mathbf{x} \in \mathbb{R}^n : \sum_i f(x_i) = 0 \}$$ have Lebesgue measure zero?

If $$f(x)$$ is a linear function, then the set is a $$n-1$$ dimensional linear subspace of $$\mathbb{R}^n$$, for which the Lebesgue measure is zero. But what if $$f(x)$$ is nonlinear?

• What methods do you know to show that a set has Lebesgue measure zero? Have you tried to find an answer for the case $n=1$ or $n=2$? – harfe May 25 at 19:35