Suppose that $\displaystyle f:[0,1]^2\longrightarrow{\mathbb{R}}$ is a function such that $\displaystyle f(.,y)$ is constant for almost all $\displaystyle y$, and $\displaystyle f(x,.)$ is constant for almost every $\displaystyle x$. Prove that $\displaystyle f$ is constant ctp (with respect to u, where u is the Lebesgue measure).

Hint: Assume the contrary. Then it sets you and you have positive measure. Use Fubini to prove that each of these sets contains at least one vertical and one horizontal interval. Conclude.

Note: A function is constant a.e., if not constant in a set of measure zero.

For each y fixed $\displaystyle f(.,y):[0,1] \times \left\{ y \right\}\longrightarrow{\mathbb{R}}$

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