Let $(X,\mathbb{X},\mu)$ be a positive measure space. Let $A$ belong to $\mathbb{X}$. If $+\infty\leq\mu(A)$ then $\mu(A)=+\infty$. Is this statement correct?
My answer is yes. I believe this is very intuitive. However, I'm not entirely convinced of the formality. Indeed, $+\infty\leq\mu(A)$ and $\mu(A)\leq +\infty$ then $\mu(A)=+\infty$. In this example, I would be treating infinity as a real number and applying the property: $a,b\in\mathbb{R}$, $a\leq b$ and $b\leq a$ then $a=b$. I would greatly appreciate your help.
This has very little to do with measure theory – it's about the axiomatic properties of the extended real numbers $\overline{\mathbb R}$ (also denoted as $\mathbb R\cup\{-\infty,\infty\}$). Since $+\infty$ is the greatest extended real number, if $x\in\overline{\mathbb R}$ and $x\ge+\infty$, then $x=+\infty$. Indeed, the axioms of the extended reals assert, among other things, that $\le$ is a total order on $\overline{\mathbb R}$, and so $\le$ has the antisymmetric property.