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YCor
YCor
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Existence of a Positive Measurable Setpositive measurable set with Disjoint Preimagedisjoint preimage under Iterated Transformationiterated transformation

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abcdmath
abcdmath
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Let $(X,\mathcal B,\mu)$ be a atomless probability measure space and $T:X\to X$ be a non-singular transformation such that $\mu\left({x\in X: T^n(x)=x}\right)=0$ for every $n\ge 1$. Let $A\in \mathcal B$ such that $\mu(A)>0$ and $k\ge 1$ a positive integer. I want to show that there exists a set $E\in \mathcal B$ such that $\mu(E)>0$ with $E\subseteq A$ and $T^{-k}(E)\cap E=\emptyset$.

Let $E=A\setminus T^{-k}(A)$ and suppose $y\in T^{-k}(E)\cap E$. Then $y\in E$ implies $y\notin T^{-k}(A)$. Also, $y\in T^{-k}(E)$ implies $T^k(y)\in E$, which further implies $T^k(y)\in A$, hence $y\in T^{-k}(A)$. This contradiction shows that $T^{-k}(E)\cap E =\emptyset$. However, I am unable to construct the set $E$ from the given hypothesis such that $\mu(E)>0$. Please help me to solve this. Thank you for your time and help.

Let $(X,\mathcal B,\mu)$ be a probability measure space and $T:X\to X$ be a non-singular transformation such that $\mu\left({x\in X: T^n(x)=x}\right)=0$ for every $n\ge 1$. Let $A\in \mathcal B$ such that $\mu(A)>0$ and $k\ge 1$ a positive integer. I want to show that there exists a set $E\in \mathcal B$ such that $\mu(E)>0$ with $E\subseteq A$ and $T^{-k}(E)\cap E=\emptyset$.

Let $E=A\setminus T^{-k}(A)$ and suppose $y\in T^{-k}(E)\cap E$. Then $y\in E$ implies $y\notin T^{-k}(A)$. Also, $y\in T^{-k}(E)$ implies $T^k(y)\in E$, which further implies $T^k(y)\in A$, hence $y\in T^{-k}(A)$. This contradiction shows that $T^{-k}(E)\cap E =\emptyset$. However, I am unable to construct the set $E$ from the given hypothesis such that $\mu(E)>0$. Please help me to solve this. Thank you for your time and help.

Let $(X,\mathcal B,\mu)$ be a atomless probability measure space and $T:X\to X$ be a non-singular transformation such that $\mu\left({x\in X: T^n(x)=x}\right)=0$ for every $n\ge 1$. Let $A\in \mathcal B$ such that $\mu(A)>0$ and $k\ge 1$ a positive integer. I want to show that there exists a set $E\in \mathcal B$ such that $\mu(E)>0$ with $E\subseteq A$ and $T^{-k}(E)\cap E=\emptyset$.

Let $E=A\setminus T^{-k}(A)$ and suppose $y\in T^{-k}(E)\cap E$. Then $y\in E$ implies $y\notin T^{-k}(A)$. Also, $y\in T^{-k}(E)$ implies $T^k(y)\in E$, which further implies $T^k(y)\in A$, hence $y\in T^{-k}(A)$. This contradiction shows that $T^{-k}(E)\cap E =\emptyset$. However, I am unable to construct the set $E$ from the given hypothesis such that $\mu(E)>0$. Please help me to solve this. Thank you for your time and help.

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abcdmath
abcdmath
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Existence of a Positive Measurable Set with Disjoint Preimage under Iterated Transformation

Let $(X,\mathcal B,\mu)$ be a probability measure space and $T:X\to X$ be a non-singular transformation such that $\mu\left({x\in X: T^n(x)=x}\right)=0$ for every $n\ge 1$. Let $A\in \mathcal B$ such that $\mu(A)>0$ and $k\ge 1$ a positive integer. I want to show that there exists a set $E\in \mathcal B$ such that $\mu(E)>0$ with $E\subseteq A$ and $T^{-k}(E)\cap E=\emptyset$.

Let $E=A\setminus T^{-k}(A)$ and suppose $y\in T^{-k}(E)\cap E$. Then $y\in E$ implies $y\notin T^{-k}(A)$. Also, $y\in T^{-k}(E)$ implies $T^k(y)\in E$, which further implies $T^k(y)\in A$, hence $y\in T^{-k}(A)$. This contradiction shows that $T^{-k}(E)\cap E =\emptyset$. However, I am unable to construct the set $E$ from the given hypothesis such that $\mu(E)>0$. Please help me to solve this. Thank you for your time and help.