Strange proposition in probability book for conditional probability I found the following proposition (15.1) in the probability book of Heinz Bauer:

Let us given that $X$ is a numeric random variable on
  $(\Omega,\mathcal{A},P)$ which is non-negative / integrable. Then to
  any sub-algebra $\mathcal{C} \subset \mathcal{A}$ it exists a almost
  surly unique non-negative / integrable random variable $X_0$, which is
  $\mathcal{C}$ measurble and it holds $$ \int_C X_0 \,dP = \int_C X \,dP
 \quad \text{for all }C\in \mathcal{C}.$$ When $X$ is integrable and
  non-negative, then $X_0$ is almost surly non-negative.

I do not get the addition at the end

When $X$ is integrable and
  non-negative, then $X_0$ is almost surly non-negative.

Isn't that already clear? Because part of the above statement is if $X \geq 0 \, \Rightarrow \, X_0 \geq 0 $? I think he is actually proving the same statement twice in one proof with different techniques - or am I missing something?
 A: I suspect what is meant where it says "non-negative/integrable" twice, it means that if $X$ is non-negative then $X_0$ is non-negative (and both might fail to be integrable) and which $X$ is integrable then $X_0$ is integrable (and both might fail to be non-negative).  If that is what was intended, then the last statement is redundant.  It's not "already clear" in the sense of requiring no proof, but it's already stated earlier in the proposition.
A: Being familiar with Bauer's notation from his book on Measure Theory, I believe Bauer wants to make the following distinction:
For a numerical (meaning extended real-valued for Bauer) non-negative (everywhere!) random variable $X$ (for such $X$ the integral always exist in $[0,\infty]$), Bauer defines the conditional expectation $X_0$ as a non-negative (everywhere!) numerical random variable which is $\mathcal{C}$ measurable and has the property
$$\int_C X_0 \,dP = \int_C X \,dP
 \quad \text{for all }C\in \mathcal{C}.$$
Such an $X_0$ exists (by the first part of the proof) and is $P$-a.s. unique, in the sense that if $X'_0$ is another non-negative (everywhere!) $\mathcal{C}$ measurable numerical random variable with the above property then $X=X_0$ $P$-a.s.
The requirement that $X_0$ be non-negative everywhere comes from the first part of the proof, which involves the existence of a density from the Radon-Nikodym theorem, and Bauer defines a density to be non-negative everywhere. Essentially this is the standard machinery in Lebesgue integration: you start with non-negative functions, and then for signed functions you consider the positive and negative parts. Hence the first part of the proof is a preliminary step for the following general case:
For a numerical (meaning extended real-valued) integrable  random variable $X$, Bauer defines the conditional expectation $X_0$ as a numerical integrable random variable which is $\mathcal{C}$ measurable and has the property
$$\int_C X_0 \,dP = \int_C X \,dP
 \quad \text{for all }C\in \mathcal{C}.$$
Such an $X_0$ exists (by the second part of the proof) and is $P$-a.s. unique, in the sense that if $X'_0$ is another $\mathcal{C}$ measurable integrable numerical random variable with the above property then $X=X_0$ $P$-a.s.
Now what about the case where $X$ is integrable and almost surely non-negative? Strictly speaking we cannot apply Bauer's first result since it concerns only everywhere non-negative random variables. This is why Bauer gives the extra argument at the end of proof.
But we can obtain this result from the first step as follows:
If $X$ is integrable and almost surely non-negative, then we can alter it, and call it say $X'$, on a null set so that $X'$ is measurable, non-negative everywhere and integrable with $\int X' dP=\int X dP$  (setting $X':=0$ on $\{X<0\}\in \mathcal{A}$ will do the job). Hence from the firt part we know there exist a non-negative (everywhere!) numerical random variable $X'_0$ which is $\mathcal{C}$ measurable and has the property
$$\int_C X'_0 \,dP = \int_C X' \,dP =\int_C X \,dP
 \quad \text{for all }C\in \mathcal{C}.$$
The special case $C=\Omega$ says that such an $X'_0$ is in addition integrable. But then from the uniqueness assertion of the second part we know that $X_0=X'_0$ $P$-a.s., that is $X_0\geq 0$ $P$-a.s.
