Why is $\infty-\infty$ undefined in measure theory?

Question

Some additions to the title: I stumbled over this problem going through my measure theory lecture notes; the author explicitly mentions that he leaves $\infty-\infty$ undefined. I would like to know what goes wrong, if I would define $l:=\infty-\infty$ for $l\in\overline{\mathbb{R}}$.

I tried to derive contradictions by playing with arithmetical rules in $\overline{\mathbb{R}}$ but couldn't obtain a contradiction.

Here's an example where I merely try to obtain a contradiction by assuming that $r:=\infty-\infty\in\mathbb{R}$ (as opposed to $\overline{\mathbb{R}}$).

\begin{eqnarray*} & \infty=\infty\\ \Rightarrow & \infty=\infty+2\\ > \Rightarrow & > \infty+\left(-\infty\right)=\left(\infty+2\right)+\left(-\infty\right) > \end{eqnarray*} an here the attempt breaks down, since this extended addition doesn't have to be associative, so one can't conclude \begin{eqnarray*} \Rightarrow & r=r+2\\ \Rightarrow & 0=2. \end{eqnarray*}

EDIT A lot of people gave me answers in which they motivated why $\infty-\infty$ doesn't make sense. This is not what I'm looking for! Motivations are nice, but only a concrete contradiction gives certainty that it absolutely makes no sense to assign a number, or $\pm \infty$ to the above expression.

Answer 1

If $l:=\infty-\infty\in \mathbb R$, then $1+l=1+(\infty-\infty)=(1+\infty)-\infty=\infty-\infty=l$, contradiction.

If $\infty-\infty=\infty$, then $-\infty=-(\infty-\infty)=-\infty+\infty = \infty-\infty=\infty$ and imilarly if $\infty-\infty=-\infty$.

So each possible choice of defining $\infty-\infty$ as a value in $\overline{\mathbb R}$ produces contradictions to the (desired) proerties of addition such as associativity and commutativity, as well as to $\infty\ne-\infty$.

You may have better luck in a one-point compactification of $\mathbb R$ where $\infty=-\infty$ holds. But that structure is unsuitable as a tool for measure theory.

Answer 2

Let's think about what situations in measure theory might be described by $\infty-\infty$. Let $A=\mathbb R,B=(0,\infty)$. Then $m(A)=m(B)=\infty$, and $$\infty-\infty\; \text{"="}\;m(A\setminus A)=0$$ yet $$\infty-\infty\; \text{"="}\;m(A\setminus B)=\infty$$ so this can't be given a consistent definition.

Answer 3

There is really no point in trying to define $\infty - \infty$ if this is going to make addition non-commutative or even non-associative. The problem is the same as defining $\frac 00$ ($\infty$ is for addition, what $0$ is for multiplication, namely an absorbing element). Also you get problems with limits (which is one of the main reasons to include $\infty$ in measure theory in the first place), because for any $x \in [-\infty ,\infty ]$ there are sequences $(a_n), (b_n)$ with $\lim_{n \to \infty }a_n=\lim_{n \to \infty }b_n=\infty$ and $\lim_{n \to \infty}(a_n-b_n)=x$, which screws up arithmetic of limits.