Finding Counterexamples for Poincaré's Recurrence Theorem (Dropped Hypothesis)

Question

I have just learned the Poincaré Recurrence Theorem and I would like to understand why each hypothesis is necessary. To this end, I am dropping one hypothesis at a time, and seeing why the theorem fails. I have found counterexample for dropping some of the hypotheses, but not all. Could you please help me with the rest?

This is the version of the theorem I have

Theorem (Poincare Recurrence): Let $f:X \to X$ where

1. $X$ is a compact metric space
2. $f$ is an m.p.t. (measure preserving transformation)
3. $f$ is continuous
4. The $\sigma$-algebra on $X$ is Borel

Then almost every point of $X$ is recurrent.

I have counterexamples for dropping $1$ and $2$, but not for $3$ and $4$.

Counterexample for 1:

$f(x) = x+1$ on $\mathbb{R}$ with Lebesgue measure. For any $x$, $f^n(x) \to \infty$, so there are no recurrent points.

Counterexample for 2: $f(x) = \frac 12 x$ on $[0, 1]$ with Lebesgue measure. Then for every $x$, $f^n(x) \to 0$, so only $x = 0$ is recurrent.