best hypothesis for data with zero mean noise is the one which assumes no noise at all?

In an ml-class, they introduced overfitting with an example:

Say, we have

• $$x$$ picked from $$Uniform(0,1)$$
• $$v$$ random noise, picked indepidently of $$x$$ from $$Uniform(-0.3,0.3)$$ and $$\mathbb{E}[v] = 0$$
• $$y = +1$$ , if $$x+v \geq \frac{1}{2}$$, otherwise $$y = -1$$

If we take a sample from this distribution, we may learn a hypothesis, such that the 0-1-loss on the sample would be zero. This hypothesis would consist of a series of intervals so that a point which falls in one of the intervals, would be classified with $$+1$$, otherwise as $$-1$$

However, an optimal hypothesis (one with minimal generalzation error) would be the one, which classifies $$x$$ as +1, if $$x < \frac{1}{2}$$, else as -1.

Intuitively I understand, that the $$\frac{1}{2}$$-hypothesis is optimal because we expect noise to be 0.

I wonder how to prove that mathematically, in particular, would the $$\frac{1}{2}$$-hypothesis stay optimal, if $$v$$ would be sampled from some other distribution with mean 0?

Any hints will be appreciated.