# Tagged Questions

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### Ergodic/stochastic convergence

I do have a problem with my homework, and to be honest I am simply lacking any idea on how to begin- maybe someone could give me a tip. First off, here is the assignment: The whole assignment deals ...
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### How can I improve this proof? On $\sum\limits^\infty f(T^nx)=\infty$ for a.a. $x \in \{f >0\}$.

Let $(X,\mathcal{F},\mu)$ be a prob. space and $T:X \rightarrow X$ be measure preserving, and $f \in \mathcal{L}^1(X,\mathcal{F},\mu)$. Assume that $f \geq 0$ and suppose the set $A = \{f > 0\}$ ...
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### $(n^2 \alpha \bmod 1)$ is equidistributed in $\mathbb{T}^2$ if $\alpha \in \mathbb{R} \setminus \mathbb{Q}$

I did the following homework question, can you tell me if I have it right? We want to show that the sequence $(n^2 \alpha \bmod 1)$ is equidistributed if $\alpha \in \mathbb{R} \setminus \mathbb{Q}$. ...
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### To construct an invariant measure on the product space

$M$ is a separable complete metric space. $N$ is a compact metric space. $F$, defined by $F(x,y)=(f(x),g(x,y))$, is a continuous transformation from $M\times N$ to itself. We know there is a measure ...
Assume that $T: X \to X$ is a homeomorphism of a compact metric space to itself, and let $\mu$ be a $T$-invariant Borel probability measure on $X$. I want to show that for almost every $x \in X$ ...
Definition: Let $X$ be a compact metric space and let $\mu$ be a Borel probability measure on $X$, then we say that the sequence $(x_n)$ of elements of $X$ is equidistributed with respect to $\mu$ ...