I was reading about Independence of sub-$\sigma$-algebras when I found the next definition:

Let $\mathcal{B}_{1},\ldots,\mathcal{B}_{n}$ $n$ sub-$\sigma$-algebras of $\mathcal{A},$ let $H$ be a subset of $[0,1],$ and let $B_{H}$ be the $\sigma-$algebra generated by $\{\mathcal{B}_{i}, i\in H\}.$ Then $\mathcal{B}_{1},\ldots,\mathcal{B}_{n}$ are said mutually independent if $\mathcal{B}_{H}$ and $\mathcal{B}_{H^{c}}$ are independent $\sigma$-algebras for every $H\in\mathcal{P}([0,1]).$

I don't understand the intuition in the definition. Why does the definition ask for $\sigma$-algebra generated by $\{\mathcal{B}_{i}, i\in H\}?$

The definition of two sub-$\sigma$-algebras $\mathcal{B}$ and $\mathcal{C}$ of $\mathcal{A}$ is $L^{2}(\mathcal{B})$ and $L^{2}(\mathcal{C})$ are orthogonal on the constant functions.

I'd appreciate any kind of idea or example.

Thanks in advanced.


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