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I am reading this book and I am a bit lost with the definitions because they are not provided and I can't seem to find it online:

Let $L_2(\Omega,A,P)$ be a probability space such that $f \in L_2$ implies that $\int_\Omega f^2 \,dP < \infty$.

The book then considers $B$ which is a sub sigma algebra of $A$.

Consider $B = \{\emptyset,\Omega\}$. The book then claims that $L_2(\Omega,B,P)$ consists of the constant functions.

My question is that: How is $L_2(\Omega,B,P)$ defined? I am confused since the definition of $L_2(\Omega,A,P)$ above doesn't seem to involve $A$. With this definition for $L_2(\Omega,B,P)$, why would it then imply that it consists of the constant functions?

Help appreciated. Thanks very much.

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    $\begingroup$ I guess the point is that a function in $L_{2}(\Omega,B,P)$ must be $B$-measurable. So that the preimage of any interval is either $\emptyset$ or all of $\Omega$. $\endgroup$ – Quinn Culver Jun 15 '14 at 22:17
  • $\begingroup$ I guess the point is that the definition was horribly reproduced. For starters, "Let $L_2(\Omega,A,P)$ be a probability space such that..." is just absurd since the probability space here is $(\Omega,A,P)$. $\endgroup$ – Did Aug 14 '14 at 10:38
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Notice that $$f:(\Omega,\{\emptyset,\Omega\})\rightarrow (\mathbb R,B(\mathbb R) )$$ where $\Omega$ is an arbitrary set is measurable if and only if $f$ is a constant function.

This is true because if $f$ is assumed to be measurable, for all $A\in B(\mathbb R)$ we have that $f^{-1}(A)\in\{\emptyset,\Omega\}$. Hence if we would assume for contradiction that $f$ is not constant then there exists at least two distinct $x_1,x_2\in\mathbb R$ such that $x_1,x_2\in f(\Omega)$ and $f^{-1}(\{x_1\})\bigcap f^{-1}(\{x_2\})=\emptyset.$ Since $x_1,x_2\in f(\Omega)$ it follows that $f^{-1}(\{x_i\})\neq \emptyset$ with $i=1,2$. Consequently since $f^{-1}(\{x_i\})\in\{\emptyset,\Omega\}$ we have that $\Omega \bigcap \Omega =f^{-1}(\{x_1\})\bigcap f^{-1}(\{x_2\})=\emptyset$. A contradicton! The other proof in the other direction works similarly.

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It seems your book is missing the elementary definitions. It is assuming you have read some previous text on measure theory.

$L_2(\Omega,\mathcal A,P)$ consists of the $\mathcal A$-meaurable functions $f$ such that $\int_\Omega |f|^2\;dP < +\infty$.

So, now you can see that the definition does, indeed, depend on $\mathcal A$.

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