You are currently browsing the category archive for the ‘FA.functional-analysis’ category.

As a quick revision, let me recollect the way we define the addition of countably many numbers:

Suppose are non-negative real numbers. Then the series is said to converge to if the sequence defined by of partial sums of converges to .

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 . Suppose is a function that associates every element of the index set to a non-negative number. Our mission is to define .

Turn into a discrete measure space by defining the measure of each point in to be 1. Clearly, now , with the discrete measure is a measure on the measure space .

Now define

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

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 be a measure space. If is such that , then except perhaps on a countable subset of I.

**Proof:**

Decompose as:

Call the right set as and the further right ones as . Suppose that some of the is infinite. Then

contrary to our assumption that the integral is finite. Hence, every is finite and so its union is countable and the proposition is proved.

Reference:

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

## Recent Comments