Linked Questions

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,...}...
101
votes
5answers
23k views

The sum of an uncountable number of positive numbers

Claim:If $(x_\alpha)_{\alpha\in A}$ is a collection of real numbers $x_\alpha\in [0,\infty]$ such that $\sum_{\alpha\in A}x_\alpha<\infty$, then $x_\alpha=0$ for all but at most countably many $\...
24
votes
3answers
4k views

Does uncountable summation, with a finite sum, ever occur in mathematics?

Obviously, “most” of the terms must cancel out with opposite algebraic sign. You can contrive examples such as the sum of the members of R being 0, but does an uncountable sum, with a finite sum, ...
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 ...
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{...
3
votes
3answers
129 views

How to prove that this set is countable?

Suppose that $(S, \Sigma)$ is a measurable space and $\mu$ is a finite measure on $(S, \Sigma)$. Suppose that whenever $x \in S$, the singleton $\{x\}$ belongs to $\Sigma$. Prove that the set $\{x \...
2
votes
3answers
122 views

Reference for series on arbitrary infinite sets

In class my professor introduced the following terminology: Let $J$ be an arbitrary infinite set and let $f:J\to\mathbb{R}$ be a real-valued function. Then the symbol $\sum_{j\in J}f(j)$ is the ...
0
votes
1answer
106 views

Concerning the definition of zero measure set

According with the book Lages Lima - Análise real, a subset $X \subset \mathbb{R}$ has zero measure whenever for every $\epsilon>0$ there is a countable cover made up of open intervals, such that ...
0
votes
2answers
62 views

Showing the Cumulative distribution only has countable plateaus

Let $F$ be a cumulative distribution function. Show that $F$ only has countably many plateaus. My idea: Define $A_{n}:=\{[a,b]\subseteq \mathbb R: [a,b]$ is a plateau of length $\geq \frac{1}{n}\}$ ...
0
votes
0answers
45 views

Defining the notion of a product over an uncountable set

Background: So a lurking thought in the back of my mind for a month or so has been the notion of a sum over an uncountable set, say $$\sum_{a \in I} a$$ for some $I$ whose cardinality is greater ...
0
votes
0answers
45 views

uncountable small set of reals [duplicate]

If the notions of "large" and "small" sets [ https://en.wikipedia.org/wiki/Large_set_(combinatorics) ] are generalized from natural numbers to positive reals, then is there an uncountable small set ...