I'm studying real analysis with Rudin Principles of Mathematical Analysis textbook.
I'm confused about the expression "the inf/sup being taken over ~~" in several definitions.
In principles of mathematical analysis Page 304, outer measure is defined as follows.
Definition $11.7$: Let $\mu$ be additive, regular, nonnegative, and finite on $\mathcal{E}$. Consider countable coverings of any set $E \subset \mathbb{R}^n$ by open elementary sets $A_n$:
$$E \subseteq \bigcup_{n=1}^\infty A_n.$$
Define
$$\mu^\ast(E) = \inf \sum_{n=1}^\infty \mu(A_n),$$
the infimum being taken over all countable coverings of $E$ by open elementary sets. $\mu^\ast(E)$ is called the outer measure of $E$, corresponding to $\mu$.
What does the "take over" mean in the expression "the infimum being taken over all countable coverings of E."?
I'm not a native English speaker. So I'm not sure what this expression exactly means.
I thought the expression is actually restricting the sets $A_n$.
Suppose E is a closed interval [0,1].
Then, $A_1 $ and $A_2$ can't be [0,2/3] and [1/3,1] because they are overlapped and the result of summation $\sum_{n=1}^\infty$ can't be infimum.
So I thought the expression "the inf being taken over all countable coverings of E by open elementary sets" is actually restricting possible $A_n$s.
Is my understanding correct?
Or could you explain the expression more detail?
Thanks!
Best Answer
It means:
The "taking" over means this: we are looking at $\inf\sum_n\mu(A_n)$, but what the hell is $A_n$? We have an inf of some quantities but we don't know where the quantities come from. So, "taken over the elementary open covers" (or words to that effect) means this (otherwise undefined!) symbol "$A_n$" refers to an element of a countable cover of $E$ by elementary open sets.
As you can see, the "fully correct" $\inf$ notation is long winded and takes up space; it's arguably nicer to describe the inf by adding some natural language and keeping the notation short.