Measure #1 « Sharing Thoughts and Knowledge

Measure #1

Posted on February 20, 2009 by ciksalma

Measure is a standard way to assign “length”, “area” and “volume”.

Now lets take length as an example. If we have an open set of $(a,b)$ . Then the length (measure) of that open set would be $\mu = b-a$ for $b>a$ .

Lets look for two open set, $(a,b)$ and $(c,d)$ . Then, the measure of the union of two open sets would be $(a,b) \cup (c,d) = (b-a)+(d-c)$ for $b>a$ and $d>c$ respectively.

We extend the example even more. How about if there is only one single point of set, $a$ . So one may find that the length would be equal to zero. We called it measure zero which is denoted by $\mu(\oslash)=0$ .

To be detailed out below is the properties of a measure $\mu$

Domain A of $\mu$ is $\sigma$ -algebra
$\mu$ is non-negative on A
$\mu$ is completely additive on A
$\mu(\oslash)=0$

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Filed under: Functional Analysis, Mathematics, Measure Theory

« Isomorphism Classes Exercise-Lebesgue Measure »

One Response

Invariant Measure « Sharing Thoughts and Knowledge, on June 4, 2010 at 12:15 pm said:

[...] the which indicates the left Haar measure and which indicates the right Haar measure. We refer to here for review on [...]

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