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Theorem 2.1 states that the set of ergodic measures is a $G_\delta$ set in the set of invariant probability measures. This means that this set is a countable intersection of open sets, this does not imply that the set is dense. Examples of $G_\delta$ sets that are not dense include the empty set, singletons etc. Theorem 3.1 in the article asserts density, but for shifts on product spaces, not for all homeomorphisms on compact spaces.

An example of a compact space on which the ergodic measures are not dense is given by $x\mapsto x^2$ on $[0,1]$ for which the invariant measures are of the form $p\delta_0 +(1-p)\delta_1$, $p\in [0,1]$, whereas the ergodic ones are just $\delta_0$ and $\delta_1$.

You are perhaps confused by the Baire Category Theorem that asserts that a countable intersection of dense open sets is dense, for example if the ambient space is a complete metric space.

Theorem 2.1 states that the set of ergodic measures is a $G_\delta$ set in the set of invariant probability measures. This means that this set is a countable intersection of open sets, this does not imply that the set is dense. Examples of $G_\delta$ sets that are not dense include the empty set, singletons etc. Theorem 3.1 in the article asserts density, but for shifts on product spaces, not for all homeomorphisms on compact spaces.

A simple example of a compact space on which the ergodic measures are not dense is given by $x\mapsto x^2$ on $[0,1]$ for which the invariant measures are of the form $p\delta_0 +(1-p)\delta_1$, $p\in [0,1]$, whereas the ergodic ones are just $\delta_0$ and $\delta_1$.

You are perhaps confused by the Baire Category Theorem that asserts that a countable intersection of dense open sets is dense, for example if the ambient space is a complete metric space.

Theorem 2.1 states that the set of ergodic measures is a $G_\delta$ set in the set of invariant probability measures. This means that this set is a countable intersection of open sets, this does not imply that the set is dense. Examples of $G_\delta$ sets that are not dense include the empty set, singletons etc. Theorem 3.1 in the article asserts density, but for shifts on product spaces, not for all homeomorphisms on compact spaces.

An example of a compact space on which the ergodic measures are not dense is given by $x\mapsto x^2$ on $[0,1]$ for which the invariant measures are of the form $p\delta_0 +(1-p)\delta_1$, $p\in [0,1]$, whereas the ergodic ones are just $\delta_0$ and $\delta_1$.

You are probably confused by the Baire Category Theorem that asserts that a countable intersection of dense open sets is dense, for example if the ambient space is a complete metric space.

Theorem 2.1 states that the set of ergodic measures is a $G_\delta$ set in the set of invariant probability measures. This means that this set is a countable intersection of open sets, this does not imply that the set is dense. Examples of $G_\delta$ sets that are not dense include the empty set, singletons etc. Theorem 3.1 in the article asserts density, but for shifts on product spaces, not for all homeomorphisms on compact spaces.

An example of a compact space on which the ergodic measures are not dense is given by $x\mapsto x^2$ on $[0,1]$ for which the invariant measures are of the form $p\delta_0 +(1-p)\delta_1$, $p\in [0,1]$, whereas the ergodic ones are just $\delta_0$ and $\delta_1$.

You are perhaps confused by the Baire Category Theorem that asserts that a countable intersection of dense open sets is dense, for example if the ambient space is a complete metric space.

Theorem 2.1 states that the set of ergodic measures is a $G_\delta$ set in the set of invariant probability measures. This means that this set is a countable intersection of open sets, this does not imply that the set is dense. Examples of $G_\delta$ sets that are not dense include the empty set, singletons etc. Theorem 3.1 in the article asserts density, but for shifts on product spaces, not for all homeomorphisms on compact spaces.

You are probably confused by the Baire Category Theorem that asserts that a countable intersection of dense open sets is dense, assuming the ambient space is a complete metric space for example.

Theorem 2.1 states that the set of ergodic measures is a $G_\delta$ set in the set of invariant probability measures. This means that this set is a countable intersection of open sets, this does not imply that the set is dense. Examples of $G_\delta$ sets that are not dense include the empty set, singletons etc. Theorem 3.1 in the article asserts density, but for shifts on product spaces, not for all homeomorphisms on compact spaces.

An example of a compact space on which the ergodic measures are not dense is given by $x\mapsto x^2$ on $[0,1]$ for which the invariant measures are of the form $p\delta_0 +(1-p)\delta_1$, $p\in [0,1]$, whereas the ergodic ones are just $\delta_0$ and $\delta_1$.

You are probably confused by the Baire Category Theorem that asserts that a countable intersection of dense open sets is dense, for example if the ambient space is a complete metric space.