I am reading Erhan Çınlar's "Probability and Stochastics". The use of linearity (underlined in red) in the following corollary stuck me.
The Fatou's Lemma would give us $\mu(\liminf (f_n-g))\le\liminf\mu(f_n-g)$ (for notational simplicity, I omitted the indicator $1_A$), which is
$\mu((\liminf f_n)-g)\le\liminf\mu(f_n-g)$.
I guess the linearity is supposed to be applied to both side to turn the above inequality into
$\mu(\liminf f_n)-\mu g\le\liminf(\mu f_n-\mu g)=\liminf\mu f_n-\mu g$
so that $\mu g$ can be cancelled to get what we proposed to prove. However, the linearity in the textbook is as follows:
From the proposition, there are two cases in which we can apply linearity correctly: 1) when two integrands are both in $\mathcal{E}_+$ and the coefficient is nonnegative. 2) when two integrands are both integrable. Looking at the left side first, since $g$ is integrable, 2) seems to be the case to apply, but in that case, $\liminf f_n$ must also be integrable. However, I could not get to this from the conditions in the corollary. If we are to use case 1), neither $\liminf f_n$ and $g$ are in $\mathcal{E}_+$, and what is worse, they are linearly combined by subtraction, not addition. So. I'm confused and don't know what kind of linearity in the proof the author means.
I tried two attempts to figure out what the linearity means.
1, I tried to prove an extended version of the linearity proposition to have $\mu((\liminf f_n)-g)=\mu (\liminf f_n)-\mu g$ using monotone convergence theorem, but I soon encountered a question: what on earth does $\mu(\liminf f_n)$ mean? The textbook defines three types of integrals:
Let's call it type a), b) and c), respectively. I can see no way to prove that $\liminf f_n\ge0$, so type b) is not applicable. I can't figure out either that $\liminf f_n$ satisfies type c). So, I'm stuck.
2, I tried to prove that $\liminf f_n$ is integrable using the domination similar to the proof of Lebesgue's dominated convergence theorem, but $f_n\ge g$ is not enough to be considered a domination that I can use to prove that $\liminf f_n$ is integrable. So I got stuck again.
I haven't yet talked about the right side of the equality, which is to show $\mu(f_n-g)=\mu f_n-\mu g$. I also have no idea how linearity can be applied to get this.
I hope anyone can help me figure out what the "linearity" mean in the proof of the corollary. I am self-studying so I don't have professor or TA to help me. It would be very good if you happen to have read this text or be using this book in a probability course. Please use the concept and notation in this book because I am just a new learner of measure-theoretical probability. Thank you.