# Sub sigma algebra and probability spaces — definition

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.

• 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 $\Omega$. – Quinn Culver Jun 15 '14 at 22:17
• 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 $(\Omega,A,P)$. – Did Aug 14 '14 at 10:38

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.
$$L_2(\Omega,\mathcal A,P)$$ 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 $$\mathcal A$$.