Linked Questions

2
votes
1answer
29 views

Meaning of absolute convergence when summing over countably infinite set.

I'm trying to understand the chapter on Elliptic functions in Stein's Complex Analysis. In particular, I am interested in the construction of Weierstrass's $\wp$ function. Let $\Lambda = \{n + m\tau :...
6
votes
1answer
124 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 ...
7
votes
1answer
581 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 ...
1
vote
1answer
135 views

Is there a rigorous way to define uncountable products?

I'm dreaming of a way to define an uncountable product of real numbers. Of course any sensible definition should only converge for a sequence with only finitely many terms outside $[0, 1]$. It ...
0
votes
0answers
45 views

Uncountably infinite series [duplicate]

I was fascinated when I heard that the most intuitive laws of arithmetic are no longer necessarily valid when it comes to the sum of an infinite sequence, which be denoted by $$S = \sum_{n=0,1,2,...}...
8
votes
2answers
288 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 ...
11
votes
4answers
827 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{...
8
votes
0answers
264 views

Would an integral defined using partitions of an interval into infinitely many intervals make sense?

In the definition of Riemann integral or Darboux-integral we first study partitions (or tagged partition) of the given interval determined by finitely many points. To each partition and a function $f$ ...
7
votes
0answers
294 views

Are uncountable “Schauder-like” bases studied/used?

We could define the following notion of basis in a way analogous to unconditional Schauder basis: If $X$ is a topological vector space over $\mathbb R$ and $B=\{b_i; i\in I\}$ be a subset of $X$. ...
0
votes
2answers
121 views

Positive Sums: Product

Lemma for: Trace Positive Sum Given the TAS $\overline{\mathbb{R}}_+$. For product sums: $$\omega:I\times J\to\overline{\mathbb{R}}_+:\quad{\sum}_{I\times J}\omega={\sum}_J{\sum}_I\omega$$ (...
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. ...
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 ...
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.
3
votes
1answer
590 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
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 $...

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