In a proof of the existence of a conditional expectation the following argument is used:
Let $\mathcal{F}$ be a sub-sigma-field of $\mathcal{G}$, wherein $(\Omega,\mathcal{G}, P)$ denotes a probability space.
Now it is claimed that $L^2(\Omega,\mathcal{F},P)$ is a (closed) subspace of $L^2(\Omega,\mathcal{G},P)$. Here is my question:
Of course an element $f$ which is square-integrable and $\mathcal{F}$-measurable is square-integrable and $\mathcal{G}$-measurable. However, the $L^2$-spaces consist of equivalence classes. And the equivalence class of $f$ are not the same: Viewed as an element in $L^2(\Omega,\mathcal{F},P)$ it consists of all $\mathcal{F}$-measurable-functions which are $a.s.$ equal to $f$. Viewed as an element in $L^2(\Omega,\mathcal{G},P)$ it consists of all $\mathcal{G}$-measurable-functions which are $a.s.$ equal to it. Hence, how can this be a subspace?
The author means that $ X = L^2(\Omega, \mathcal{F}, P) $ is a subspace of $ Y = L^2(\Omega, \mathcal{G}, P) $ in the sense that there is a natural inclusion $ \iota : X \hookrightarrow Y $. Let $ \tilde{X} $ denote the full set of square-integrable $ \mathcal{F} $-measurable functions so that $ X = \tilde{X} / \sim $ where $ \sim $ is the a.e. equivalence relation on $ \mathcal{F} $-measurable functions. An important observation is that $ \sim $ is just the restriction of a.e. equivalence on $ \mathcal{G} $-measurable functions to $ \mathcal{F} $-measurable functions.
So if we define $ \tilde{\iota} : \tilde{X} \rightarrow \tilde{Y} $ by actual inclusion (every $ \mathcal{F}$-measurable function is indeed a $ \mathcal{G} $-measurable function) then $ \tilde{\iota} $ gives rise to a map $ \iota : X \rightarrow Y $. Specifically, $ \iota $ is the unique map making the diagram commute $\require{AMScd}$ \begin{CD} \tilde{X} @>{\tilde{\iota}}>> \tilde{Y} \\ @VVV @VVV\\ X @>{\iota}>> Y \end{CD} where the vertical arrows are the quotient maps. Due to our observation about these quotient maps being "compatible", $ \iota $ is well-defined. The inclusion $ \iota $ is what the author means when they say that $ X $ is a subspace of $ Y $.