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As a quick revision, let me recollect the way we define the addition of countably many numbers:
Suppose $a_0, a_1, \dots$ are non-negative real numbers. Then the series $a_1 + a_2 + \dots$ is said to converge to $L$ if the sequence $s_n$ defined by $s_n = a_1 + a_2 + \dots + a_n$ of partial sums of $a_n$ converges to $L$.

In other words, the tail of the sequence must add up to atmost as small a number as one desires.

Now, in the quest of adding up uncountably many non-negative numbers, let us index the set of these numbers by $I$. Suppose $f : I \to [0, \infty)$ is a function that associates every element of the index set $I$ to a non-negative number. Our mission is to define $\sum_{x \in I} f(x)$.

Turn $I$ into a discrete measure space by defining the measure of each point in $I$ to be 1. Clearly, now $I$, with the discrete measure $\mu : \mathcal P (I) \to {1}$ is a measure on the measure space $I$.

Now define

$\sum_{ x \in I} f(x)= \int_I f(t) d\mu (t)$

In other words, the sum of all elements indexed by $I$, is the integral of f with respect to the discrete measure $\mu$ taken over the entire index set $I$.

I then wondered if the sum of uncountably many positive numbers could be finite. (Note that a countable sum of positive numbers can be finite; the reciprocals of positive powers of two add up to unity.) The following proposition helps (cf. [Rud]) :

Proposition: Let $(I, \mathcal P (I), \mu : I \to \{1\} )$ be a measure space. If $f : I \to [0,\infty)$ is such that $\int_I f(t) d\mu(t)<\infty$, then $f\equiv 0$ except perhaps on a countable subset of I.

Proof:
Decompose $I$ as:
$I = \{ x | f(x) = 0 \} \bigcup_{n \in N} \{ x | f(x) > 1/n \}$
Call the right set as $A_0$ and the further right ones as $A_n$. Suppose that some of the $A_n$ is infinite. Then
$\int_I f(t) d\mu(t) > \int_{A_n} \frac{1}{n} d\mu(t) = \infty$
contrary to our assumption that the integral is finite. Hence, every $A_n$ is finite and so its union is countable and the proposition is proved.

Reference:
[Rud] Walter Rudin, “Real and Complex Analysis”, 3rd ed. TMH

Abhishek Parab

I? An Indian. A mathematics student. A former engineer. A rubik's cube addict. A nature photographer. A Pink Floyd fan. An ardent lover of Chess & Counter-Strike.

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