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Is the average of all $x$ such that $0 \le x \le 1$ $ \forall $ $x \in \mathbb{R}$ equal to $\frac{1}{2}$ ?

It may seem this way but how can we show that indeed it is $\frac{1}{2}$. We know that the number of elements in $0$ to $1$ of real numbers is $\aleph_1$. therefore we have uncountable over uncountable..? Is this true or not? thanks..

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You cannot really average the elements, but you can take expectations with respect to the uniform distribution, and this expectation is indeed $1/2$. You obtain it as the integral $\int_0^1 x~dx=1/2$. –  Michael Greinecker Mar 28 '12 at 7:15
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If you think from a probability perspective, i.e. $x$ is a uniform distribution over (0,1), then it makes sense. –  Hawii Mar 28 '12 at 7:19
    
So this means that it is impossible to find the average of any uncountable number of elements? and what could be the distinction between the average point-of-view and the uniform distribution? –  Keneth Adrian Mar 28 '12 at 7:32
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"We know that the number ... is $\aleph_1$." No, this is (CH). We know that it is $2^{\aleph_0}$. –  martini Mar 28 '12 at 7:59
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@Keneth Adrian: Whether that is true cannot be decide by the usual axioms of set theory. Take a look at the continuum hypothesis. –  Michael Greinecker Mar 28 '12 at 8:15

1 Answer 1

up vote 2 down vote accepted

If $F$ is a finite set and $f:F\to\mathbb{R}$ is a function, we can take its average as $1/N \sum_{x\in F}f(x)$, where $N$ is the number of elements of $F$. The average of a finite set $F$ of real numbers is simply the average of the identity on $F$.

If we want to extend this to a nonnegative function $f:X\to\mathbb{R}$ with $X$ an infinite set, we encounter two problems. First, since the cardinality of $X$ is not a real number, it is not clear what dividing by that number means. Second, if $X$ is uncountable and $f(x)\neq 0$ for uncountably many $x$, the sum $\sum_{x\in X}f(x)$ is necessarily infinite. If $f$ is not nonnegative, we can in addition get problems with convergence of the sum.

In order to solve the first problem, we observe that we can sometimes use a different notion of size for the underlying set $N$. For eaxample, a closed interval $[a,b]$ is necessarily uncountable if $a<b$, but its length $b-a$ might be a good notion of it size for our purposes. By extending this notion of lenght to more complicated sets of real numbers (but not to all sets of real number!) we arrive at the notion of Lebesgue measure.

To deal with the second problem, we can ask ourselves what a good alternative to adding up might be. If $f$ is constant with value $c$ on the interval $[a,b]$ and zero everywhere else, we would want that "replacement of sum"$/$"size of $[a,b]$"$=$ "replacement of sum"$/(b-a)=c$. So "replacement of sum"$=c(b-a)$. Now the sum of the values of $f+g$ is in the finite case the sum of the values of $f$ plus the sum of the values of $g$. Also, the sum of the values of $\alpha f$ equals the sum of the values of $f$ times $\alpha$ when $\alpha$ is a real number. So we can sum up functions that are linear combinations of "blocks". Extending this by a limiting operation to a wider class of functions gives us the integral as a replacement of summation.

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