Suppose $\varphi:\mathbb CP^1\to \mathbb CP^1$ is a homeomorphism that is holomorphic on a connected open subset $U\subset \mathbb CP^1$ such that $\mathbb CP^1\setminus U$ has zero measure.

Is it true that $\varphi$ is holomorphic on the whole $\mathbb CP^1$ (so it is a projective transformation)?

If no, what kind of assumptions of $U$ would suffice? (for example $\mathbb CP^1\setminus U$ has Hausdorff dimension $\le 1$?)

Suppose $\varphi:\mathbb CP^1\to \mathbb CP^1$ is a homeomorphism holomorphic on a connected open subset $U\subset \mathbb CP^1$ with $\mathbb CP^1\setminus U$ of zero measure.

Is it true that $\varphi$ is holomorphic on the whole $\mathbb CP^1$ (so it is a projective transformation)?

If no, what kind of assumptions of $U$ would suffice? (for example $\mathbb CP^1\setminus U$ has Hausdorff dimension $\le 1$?)