  Let $M$ be a compact 1-manifold. Given a continuous function $g:M\to \mathbb
R_+$ we consider the following ordinary differential equation:
$\|\dot{f}(t)\|=g(t)$, where $f:M\to \mathbb R^2$. We construct a probability
measure on the space of almost everywhere differentiable solutions of this
differential equation and study this measure. A solution of this equation can
be viewed as an isometric immersion of a compact 1-manifold into $\mathbb R^2$.
Nash's convergence technique in the proof of isometric $C^1$-immersion theorem
plays an important role in the construction.
