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.