Linked Questions

6
votes
1answer
121 views

Is there a sum of an uncountable set of Real numbers? [duplicate]

The addition of Real numbers is commutative, so instead of saying we can find the sum of a sequence $\{a_1,...,a_n\}$ of real numbers that are pairwise not equal, we can say that there is a sum of a ...
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 $...
28
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
825 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{...
7
votes
2answers
811 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 ...
15
votes
1answer
475 views

A question about $\prod_{x\in \mathbb{R}^{*}}{x}$

When I was an elementary school student, I encountered some puzzles like "What's product of numbers of hair of each people in the world?", which answer was zero, because there's people with no hair(...
1
vote
2answers
1k views

What is the sum of all real numbers from $0$ to $1$? [closed]

I wanted to know the approximate sum of real numbers from 0 to 1. Please tell me how we can find it.
9
votes
1answer
871 views

Transfinite series: Uncountable sums

If you sum an expression over an uncountable set $\sum_{x\in \mathbb{R}}f(x)$, then do we need $f(x)=0$ on all but a countable subset in order for the sum to have a finite value? If not can you give ...
3
votes
1answer
583 views

Is every Hilbert space an $L^2$ space?

Let $H$ be any Hilbert space. Must there exist a measure space $(X,\scr{M},\mu)$ such that we have a Hilbert space isomorphism: $$H\cong L^2(\mu)$$ Thank you
7
votes
2answers
278 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 ...
2
votes
4answers
261 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
300 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 ...
1
vote
2answers
338 views

A question about an uncountable summation. [duplicate]

Possible Duplicate: The sum of an uncountable number of positive numbers Consider $\sum_{\lambda \in \Lambda} a_{\lambda}$ . Here all $a_\lambda $ is non-negative. Then I want to prove that if $\...
7
votes
1answer
557 views

Is Lebesgue integral w.r.t. counting measure the same thing as sum (on an arbitrary set)?

TL:DR; For arbitrary sets (not necessarily countable) we have two notions: Lebesgue integral w.r.t. the counting measure and sum of family indexed by this set. Are these two notions equivalent? For ...

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