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I have a curiosity on the Ergodic decomposition given by the von Neumann's theorem:

$$L^2(X,\Sigma,\mu)=L^2(X,\Sigma_T,\mu)\oplus\overline{\{f-f\circ T\ :\ f\in L^2(X,\Sigma,\mu)\}},$$

that occurs for a measure-preserving map $T$ of a probability space $(X,\Sigma,\mu)$, $\Sigma_T$ being the sub-σ-algebra of all $T$-invariant measurable sets, and the (orthogonal) projector being the conditional expectation $E(\cdot|\Sigma_T)$. For all $1\leq p \leq\infty$, the conditional expectation is well-defined as a linear projector of norm 1 on $\textstyle L^p(X,\Sigma,\mu)$, with range the closed subspace $L^p(X,\Sigma_T,\mu) \subset L^p(X,\Sigma,\mu)$. Therefore it's quite natural to consider the analogue decomposition of the $L^p$ spaces given by the $L^p$ projector , that ''should'' be:

$$L^p(X,\Sigma,\mu)=L^p(X,\Sigma_T,\mu)\oplus\ {\overline{\{ f-f\circ T\ :\ f\in L^p(X,\Sigma,\mu)}\} }^{L^p}.$$

Now if $1\leq p\leq 2$, this splitting actually holds true, and it is easily obtained with a Lp-closure starting from the $L^2$ splitting. For $2\leq p \leq\infty$, a splitting is obtained by restriction to $L^p(X,\Sigma,\mu)$. And here it comes the problem: this way I get the right first factor (the range of the projector) $L^p(X,\Sigma_T,\mu)=L^2(X,\Sigma_T,\mu)\cap L^p(X,\Sigma,\mu)$, but I can't see why the kernel of the projector,

$$\overline{ \{f-f\circ T\ :\ f\in L^2(X,\Sigma,\mu) \} }^{L^2} \cap\ L^p(X,\Sigma,\mu)$$

should be equal to (and not larger than)

$$\overline{\{f-f\circ T\ :\ f\in L^p(X,\Sigma,\mu)\}}^{L^p}.$$

Maybe it's not a fundamental point (it does not enter in the proof of the main ergodic theorems) but I think that if a complete analogy holds true, it would be nice to state it, and if it doesn't, one would like to know what goes wrong. I checked the main texts of ergodic theory on this point, and found nothing.

Summarizing:

is there a (hopefully quick) way to see whether for $2\leq p \leq\infty$ there is an inclusion (hence equality)

$$\overline{ \{f-f\circ T\ :\ f\in L^2(X,\Sigma,\mu) \} }^{L^2} \cap\ L^p(X,\Sigma,\mu)\ \subset \ \overline{\{f-f\circ T\ :\ f\in L^p(X,\Sigma,\mu)\}}^{L^p}$$

$$\mathbf{?}$$

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I have a curiosity on the Ergodic decomposition given by the von Neumann's theorem:

$$L^2(X,\Sigma,\mu)=L^2(X,\Sigma_T,\mu)\oplus\overline{\{f-f\circ T\ :\ f\in L^2(X,\Sigma,\mu)\}},$$

that occurs for a measure-preserving map $T$ of a probability space $(X,\Sigma,\mu)$, $\Sigma_T$ being the sub-σ-algebra of all $T$-invariant measurable sets, and the (orthogonal) projector being the conditional expectation $E(\cdot|\Sigma_T)$. For all $1\leq p \leq\infty$, the conditional expectation is well-defined as a linear projector of norm 1 on $\textstyle L^p(X,\Sigma,\mu)$, with range the closed subspace $L^p(X,\Sigma_T,\mu) \subset L^p(X,\Sigma,\mu)$. Therefore it's quite natural to consider the analogue decomposition of the $L^p$ spaces given by the $L^p$ projector , that ''should'' be:

$$L^p(X,\Sigma,\mu)=L^p(X,\Sigma_T,\mu)\oplus\ {\overline{\{ f-f\circ T\ :\ f\in L^p(X,\Sigma,\mu)}\} }^{L^p}.$$

Now if $1\leq p\leq 2$, this splitting actually holds true, and it is easily obtained with a Lp-closure starting from the $L^2$ splitting. For $2\leq p \leq\infty$, a splitting is obtained by restriction to $L^p(X,\Sigma,\mu)$. And here it comes the problem: this way I get the right first factor (the range of the projector) $L^p(X,\Sigma_T,\mu)=L^2(X,\Sigma_T,\mu)\cap L^p(X,\Sigma,\mu)$, but I can't see why the kernel of the projector,

$$\overline{ \{f-f\circ T\ :\ f\in L^2(X,\Sigma,\mu) \} }^{L^2} \cap\ L^p(X,\Sigma,\mu)$$

should be equal to (and not larger than)

$$\overline{\{f-f\circ T\ :\ f\in L^p(X,\Sigma,\mu)\}}^{L^p}.$$

Maybe it's not a fundamental point (it does not enter in the proof of the main ergodic theorems) but I think that if a complete analogy holds true, it would be nice to state it, and if it doesn't, one would like to know what goes wrong. I checked the main texts of ergodic theory on this point, and found nothing.

Summarizing:

is there a (hopefully quick) way to see whether for $2\leq p \leq\infty$ there is an inclusion

$$\overline{ \{f-f\circ T\ :\ f\in L^2(X,\Sigma,\mu) \} }^{L^2} \cap\ L^p(X,\Sigma,\mu)\ \subset \ \overline{\{f-f\circ T\ :\ f\in L^p(X,\Sigma,\mu)\}}^{L^p}$$

$$\mathbf{?}$$

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# Ergodic splitting in L<sup>p</sup>L_p

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