What is a Measure? (Measure Theory)

A person can have ‘Measure’ in Internet’s Big Data sense:
A man’s Self = Measure = ∪ {body, psychic, clothes, house, wife, children, ancestors, friends, reputation, job, car, bank-account, websites visited, online buying pattern, hobbies, interests, lifestyle, …}

You become your algorithmic self: identity and identification shifted to an entirely digital (therefore measurable) plane.

Google, Amazon, etc … collect your quantified “Measure”. Not only they amass large amount of data about you, but also use algorithms to make sense of these data.

Your ‘Measure’ is a big business in the age of Big Data, or ‘DT’ (Data Technology) Age — as coined by Jack Ma of Alibaba.com.


Singapore Maths Tuition

In layman’s terms, “measures” are functions that are intended to represent ideas of length, area, mass, etc. The inputs for the measure functions would be sets, and the output would be a real value, possibly including infinity.

It would be desirable to attach the value 0 to the empty set $latex emptyset$ and measures should be additive over disjoint sets in X.

Definition (from Bartle): A measure is an extended real-valued function $latex mu$ defined on a $latex sigma$-algebra X of subsets of X such that
(i) $latex mu (emptyset)=0$
(ii) $latex mu (E) geq 0$ for all $latex Ein mathbf{X}$
(iii) $latex mu$ is countably additive in the sense that if $latex (E_n)$ is any disjoint sequence ($latex E_n cap E_m =emptyset text{if }nneq m$) of sets in X, then

$latex displaystyle mu(bigcup_{n=1}^infty E_n )=sum_{n=1}^infty mu (E_n)$.

If a measure does not take on $latex +infty$, we say…

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