I am reading "Measure, Integration & Real Analysis" by Sheldon Axler.
3.8 integration is order preserving
Suppose $(X,\mathcal{S},\mu)$ is a measure space and $f,g:X\to [0,\infty]$ are $\mathcal{S}$-measurable functions such that $f(x)\leq g(x)$ for all $x\in X$. Then $\int fd\mu\leq\int gd\mu.$
3.20 integration is homogeneous
Suppose $(X,\mathcal{S},\mu)$ is a measure space and $f:X\to [-\infty,\infty]$ is a function such that $\int fd\mu$ is defined. If $c\in\mathbb{R}$, then $$\int cfd\mu=c\int fd\mu.$$
3.21 additivity of integration
Suppose $(X,\mathcal{S},\mu)$ is a measure space and $f,g:X\to\mathbb{R}$ are $\mathcal{S}$-measurable functions such that $\int |f|d\mu<\infty$ and $\int |g|d\mu<\infty$. Then $$\int (f+g)d\mu=\int f d\mu+\int g d\mu.$$
3.22 integration is order preserving
Suppose $(X,\mathcal{S},\mu)$ is a measure space and $f,g:X\to\mathbb{R}$ are $\mathcal{S}$-measurable functions such that $\int fd\mu$ and $\int gd\mu$ are defined. Suppose also that $f(x)\leq g(x)$ for all $x\in X$. Then $\int fd\mu\leq\int gd\mu.$
Proof The cases where $\int fd\mu=\pm\infty$ or $\int gd\mu=\pm\infty$ are left to the reader.
$\dots$
I proved the following proposition.
Is my proof ok?
Proposition:
Suppose $(X,\mathcal{S},\mu)$ is a measure space and $f,g:X\to\mathbb{R}$ are $\mathcal{S}$-measurable functions such that $\int fd\mu$ and $\int gd\mu$ are defined. Suppose also that $f(x)\leq g(x)$ for all $x\in X$. Suppose $\int fd\mu=\pm\infty$ or $\int gd\mu=\pm\infty$. Then $\int fd\mu\leq\int gd\mu.$
My proof:
If $\int fd\mu=-\infty$, then obviously $\int fd\mu\leq\int gd\mu.$
If $\int gd\mu=+\infty$, then obviously $\int fd\mu\leq\int gd\mu.$Let $\int fd\mu=+\infty$.
Then, $+\infty=\int fd\mu=\int f^+d\mu-\int f^-d\mu$.
So, $\int f^+d\mu=+\infty$ and $\int f^-d\mu\in\mathbb{R}$.
$f^+(x)-f^-(x)=f(x)\leq g(x)$ holds.
$f^+(x)\leq g(x)+f^-(x)$ holds.
By 3.8, $\int f^+d\mu\leq \int \left(g+f^-\right)d\mu$ holds.
By 3.21, $\int f^+d\mu\leq \int gd\mu+\int f^-d\mu$ holds.
Since $\int f^-d\mu\in\mathbb{R}$, the following inequality is not nonsense and holds:
$\int fd\mu=\int f^+d\mu-\int f^-d\mu\leq \int gd\mu$.Let $\int gd\mu=-\infty$.
Then, $-\infty=\int gd\mu=\int g^+d\mu-\int g^-d\mu$.
So, $\int g^+d\mu\in\mathbb{R}$ and $\int g^-d\mu=+\infty$.
$f(x)\leq g(x)=g^+(x)-g^-(x)$ holds.
$g^-(x)\leq g^+(x)-f(x)$ holds.
By 3.8, $\int g^-d\mu\leq \int (g^+-f)d\mu$ holds.
By 3.20 and 3.21, $\int g^-d\mu\leq \int g^+d\mu-\int fd\mu$ holds.
Since $\int g^-d\mu=+\infty$, $\int g^+d\mu-\int fd\mu=+\infty$.
Since $\int g^+d\mu\in\mathbb{R}$, $\int fd\mu=-\infty$.
So, $\int fd\mu=\int gd\mu.$