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Let $\mathcal{L} \subset \mathbb{R}$ the Lebesgue sigma algebra and $\mathcal{B} \subset \mathbb{R}^{n}$ the Borel sigma algebra. I'll denotes by $\mathcal{L} \times \mathcal{B}$ the smallest sigma algebra containing products $A \times B$, where $A \in \mathcal{L}$ and $B \in \mathcal{B}$.

Let $\mathbb{T} \subset \mathbb{R}$ a compact set and $\Delta$ a sigma algebra of subsets of $\mathbb{T}$. Furthermore, if $E \in \Delta$ then $E \in \mathcal{L}$. If $E \subset \mathbb{T}$ and $E \in \mathcal{L}$ then $E \in \Delta$.

If $D \subset \mathbb{T} \times \mathbb{R}^{n}$ and $D \in \mathcal{L} \times \mathcal{B}$ then $D \in \Delta \times \mathcal{B}$ ?

Thanks.

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Since $\mathbb{T}$ is Borel, doesn't your second paragraph just say that $\Delta$ is the restriction of $\mathcal{L}$ to $\mathbb{T}$? Wouldn't it then be the case that $\Delta\times\mathcal{B}$ is the restriction of $\mathcal{L}\times\mathcal{B}$ to $\mathbb{T}\times\mathbb{R}^n$? –  François G. Dorais Feb 12 '12 at 14:23
    
Dear Santos, since you have asked several questions on measure theory and had each one closed, perhaps you could give some context or some indication of why you want to know the answer. Currently this series of bald questions gives an unfortunate impression of someone trying to get their coursework done for them. If that's not the case, then why not add some words as to how the question came up, or which parts you can already do, etc. –  Yemon Choi Feb 12 '12 at 18:48
1  
I'm finishing a work on the control on time scales (don't is homework). The sigma algebra product is common in control theory. But I have some doubts about it. –  Santos Feb 13 '12 at 0:57

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