Proof that sum, product, min, max of two measurable functions is measurable $(M, F)$ measurable space. 
Let $f,g$ be positive measurable functions $f,g : M \to [0, \infty ]$.
I need to show that their sum, product, min and max are measurable.
I have seen this question already, but it is always for the functions that map to $\mathbb{R}$. Now I am interested to know this for positive measurable functions. 
Thank you very much for any of your help.
 A: Define $\hat f = f\mathbf{1}_{\{f<\infty\}}$ and similarly define $\hat{g}$. Notice that both $\hat f$ and $\hat g$ are measurable $M\to \mathbb{R}$. The functions (1) $f+g$, (2) $fg$, (3) $\min(f,g)$ and (4) $\max(f,g)$ will coincide with (1) $\hat f + \hat g$, (2) $\hat f \hat g$, (3) $\min \left( \hat f, \hat g \right)$ and (4) $\max \left( \hat f, \hat g \right)$ unless (1) $f$ or $g$ is infinite, (2) $f = \infty, g \neq 0$ or $f \neq 0, g = \infty$, (3) $f = g = \infty$, (4) $f$ or $g$ is infinite and is these later cases the functions of $f$ and $g$ have the value infinity. Therefore $f + g = \hat f + \hat g + \infty \mathbf{1}_{\{f=\infty\}\cup\{g=\infty\}}$, $fg = \hat f \hat g + \infty \mathbf{1}_{\{f=\infty, g \neq 0\}\cup\{f\neq0,g=\infty\}}$, $\min(f,g) = \min \left( \hat f, \hat g \right) + \infty \mathbf{1}_{\{f=\infty,g=\infty\}}$, $\max(f,g) = \max \left( \hat f, \hat g \right) + \infty \mathbf{1}_{\{f=\infty\}\cup\{g=\infty\}}$. Henceforth, all the functions of $f$ and $g$ are measurable.
