I am reading a paper on task allocation
Definition 3
Let j ∈ {1, …, Nt}.
Then the utility of a task tj is defined as a function
Uj:R+∪{∞}→R+ such that Uj(∞) = 0.
I am trying to understand how to read the infinity part and what does union with infinity mean
So its the function Uj maps positive real number union with infinity to positive real number such as Uj with infinity domain equal 0
Is that correct way to read and how to interpret it? The following is the reference article.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5261615/#pone.0170659.e020
Properly typeset, as in the article, it is
$U_j$ is a function from $\Bbb R^+ \cup \{\infty\}$ to $\Bbb R^+$ such that $U_j(\infty)=0$
$\Bbb R^+$ is the set of positive real numbers. The domain of the function is all the positive numbers plus infinity, which we sometimes call the extended positive reals. It just adds one more point to the end of the line, which we call infinity and is greater than any other number. Each $U_j$ is a function from this set to the positive reals. Probably these functions are decreasing, but we are not told that at this point. Then we specify that the value of the function is $0$ when the argument is $\infty$. There is a glitch in the definition because the range is listed as $\Bbb R^+$, but the value at $\infty$ is $0$, which is not in $\Bbb R^+$. Maybe $\Bbb R^+$ is intended to include $0$, or maybe the value of the function must be positive at all real inputs but is $0$ at infinity.
Sometimes using the extended reals lets you avoid talking about limits at infinity. I haven't seen a case where the extended reals were necessary. One could recast the argument to work in the standard reals by using limits.