If $f \circ V=f$ implies $f$ is constant, then $V$ must be ergodic.

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Hint: What happens if $f = \chi_A$ is the characteristic function of a set $A$? What does it mean for $\chi_A \circ V = \chi_A$? What does it mean for $\chi_A$ to be constant a.e?

To follow up with your question in the comments, it's a kind of "standard trick" in measure theory. If you know things about sets, then we can often pass to functions by first considering characteristic functions, then simple functions (by taking linear combinations), then all measurable functions (by taking limits). For instance, this is how we define the Lebesgue integral.

Conversely, if we know things about functions, then we can often recover information about sets by seeing what happens to characteristic functions. Notice this is the "easier" direction of the two, because functions are the more complicated object. That said, this trick of considering characteristic functions is still extremely useful!

I hope this helps ^_^