Linked Questions

20
votes
6answers
2k views

Is there a number that's right in the middle of this interval $(0, 1)$?

This might seem like a silly question, but is there a number that's right in the middle of this interval $(0, 1)$? And the half-open intervals: $(0, 1]$, $[0, 1)$? I know for a fully closed interval $...
29
votes
5answers
6k views

use of $\sum $ for uncountable indexing set

I was wondering whether it makes sense to use the $\sum $ notation for uncountable indexing sets. For example it seems to me it would not make sense to say $$ \sum_{a \in A} a \quad \text{where A is ...
18
votes
4answers
3k views

Can we add an uncountable number of positive elements, and can this sum be finite?

Can we add an uncountable number of positive elements, and can this sum be finite? I always have trouble understanding mathematical operations when dealing with an uncountable number of elements. ...
11
votes
4answers
833 views

Why “countability” in definition of Lebesgue measures?

According to Wikipedia, the definition of the Lebesgue outer measure of a set $E$ is as follows: $$ \lambda^*(E) = \operatorname{inf} \left\{\sum_{k=1}^\infty l(I_k) : {(I_k)_{k \in \mathbb N}} \text{...
2
votes
4answers
263 views

How to find the limit of an uncountably infinite expression?

For example: What is the sum of all positive integers divided by the square of the amount of positive integers? $$\frac{1+2+3+4+5+6+\cdots}{(1+1+1+1+1+1+\cdots)^2}$$ Then the limit is \begin{align*}...
3
votes
3answers
301 views

What is this sum? [duplicate]

Possible Duplicate: The sum of an uncountable number of positive numbers Consider the following question: For each real number $x$, let $\epsilon_x>0$ be an associated positive number. Is the ...
2
votes
3answers
420 views

Can absolute convergent series be expressed as sum of two series?

Let $C\subset \omega \bigwedge A\bigcap B = \emptyset \bigwedge A\bigcup B = C$. Let $\{x_i\}$ be a sequence of nonnegative reals. Suppose $C$ is infinite and $\sum_{i\in C} x_i$ converges. (Since ...
8
votes
2answers
300 views

Sum of a series indexed by ordinals

If $\mu$ is an ordinal, how can we formalize that $$ \sum_{\lambda<\mu}x_{\lambda}=z $$ When $\mu=\omega$, this is just the usual infinite series, the partial sums converge to $z$. What is the ...
7
votes
2answers
812 views

Why do we distinguish between infinite cardinalities but not between infinite values?

More specifically, why are we "allowed" to denote $|\mathbb{N}|<|\mathbb{R}|$ but not $\sum\limits_{n\in\mathbb{N}}1<\sum\limits_{r\in\mathbb{R}}1$? Can we distinguish between "countable ...

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