Measure

Main.Measure History

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February 22, 2007, at 04:06 PM by Kyriakos -
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November 23, 2006, at 03:27 PM by Kyriakos -
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Consider a non-empty set \fs4 \Omega. Also consider a \fs4 \sigma-algebra on \fs4 \Omega, denoted with \fs4 \cal{F}. The pair \fs4 (\Omega, \cal{F}) is called a measure space, since we can apply the measure function on the members of \fs4 \cal{F}. The members of \fs4 \cal{F} are called just measurable.

Contrary to intuition, not all subsets of \fs4 \Omega are (necessarily) measurable, and therefore there are subsets that don't belong in any \fs4 \sigma-algebra \fs4 \cal{F}. An important example of such subsets is the Vitali set.

to:

Consider a non-empty set \fs4 \Omega. Also consider a \fs4 \sigma-algebra on \fs4 \Omega, denoted with \fs4 \mathscr{F}. The pair \fs4 (\Omega, \mathscr{F}) is called a measure space, since we can apply the measure function on the members of \fs4 \mathscr{F}. The members of \fs4 \mathscr{F} are called just measurable.

Contrary to intuition, not all subsets of \fs4 \Omega are (necessarily) measurable, and therefore there are subsets that don't belong in any \fs4 \sigma-algebra \fs4 \mathscr{F}. An important example of such subsets is the Vitali set.

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\fs4 \mu : \cal{F} \rightarrow [0, +\infty]
to:
\fs4 \mu : \mathscr{F} \rightarrow [0, +\infty]
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  • The measure of a countable union of disjoint sets equals the sum of their measures. If \fs4 F_1, F_2, \ldots \in \cal{F} with \fs4 F_i \cap F_j = \emptyset for \fs4 i \neq j, then
to:
  • The measure of a countable union of disjoint sets equals the sum of their measures. If \fs4 F_1, F_2, \ldots \in \mathscr{F} with \fs4 F_i \cap F_j = \emptyset for \fs4 i \neq j, then
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If we look at the members of \fs4 \cal{F} as events then we can view this measure as the probability of these events, and therefore we call this measure a probability measure. Equipped with this probability measure, the measure space is called a probability space.

to:

If we look at the members of \fs4 \mathscr{F} as events then we can view this measure as the probability of these events, and therefore we call this measure a probability measure. Equipped with this probability measure, the measure space is called a probability space.

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November 23, 2006, at 03:25 PM by Kyriakos -
Changed lines 24-28 from:

If we look at the members of \fs4 \cal{F} as events then we can view this measure as the probability of these events, and therefore we call this measure a probability measure. Equipped with this probability measure, the measure space is called a probability space.

to:

If we look at the members of \fs4 \cal{F} as events then we can view this measure as the probability of these events, and therefore we call this measure a probability measure. Equipped with this probability measure, the measure space is called a probability space.

What links here? (:pagelist link={$FullName} fmt=simple:)

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November 03, 2006, at 01:05 AM by Kyriakos -
Changed lines 1-4 from:

The measure is a mathematical concept that generalizes the everyday notion of size. Intuitively one can think of measure as length, weight, volume, distance from a point, probability, etc.

Consider a non-empty set \fs4 \Omega. Also consider a \fs4 \sigma-algebra on \fs4 \Omega, denoted with \fs4 \cal{F}. The pair \fs4 (\Omega, \cal{F}) is called a measurable space, since we can apply the measure function on the members of \fs4 \cal{F}. The members of \fs4 \cal{F} are called just measurable.

to:

Measure is a mathematical concept that generalizes the everyday notion of size. Intuitively one can think of measure as length, weight, volume, distance from a point, probability, etc.

Consider a non-empty set \fs4 \Omega. Also consider a \fs4 \sigma-algebra on \fs4 \Omega, denoted with \fs4 \cal{F}. The pair \fs4 (\Omega, \cal{F}) is called a measure space, since we can apply the measure function on the members of \fs4 \cal{F}. The members of \fs4 \cal{F} are called just measurable.

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If we look at the members of \fs4 \cal{F} as events then we can view this measure as the probability of these events, and therefore we call this measure a probability measure. Equipped with this probability measure, the measure space is called a probability space.

to:

If we look at the members of \fs4 \cal{F} as events then we can view this measure as the probability of these events, and therefore we call this measure a probability measure. Equipped with this probability measure, the measure space is called a probability space.

Restore
November 03, 2006, at 01:02 AM by Kyriakos -
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  1. The empty set has measure zero
to:
  • The empty set has measure zero
Changed lines 16-17 from:
  1. The measure of a countable union of disjoint sets equals the sum of their measures. If \fs4 F_1, F_2, \ldots \in \cal{F} with \fs4 F_i \cap F_j = \emptyset for \fs4 i \neq j, then
to:
  • The measure of a countable union of disjoint sets equals the sum of their measures. If \fs4 F_1, F_2, \ldots \in \cal{F} with \fs4 F_i \cap F_j = \emptyset for \fs4 i \neq j, then
Restore
November 03, 2006, at 01:02 AM by Kyriakos -
Changed lines 1-2 from:

Consider a non-empty set \fs4 \Omega. Also consider a \fs4 \sigma-algebra on \fs4 \Omega, denoted with \fs4 \cal{F}. A measure \fs4 \mu will be a function

to:

The measure is a mathematical concept that generalizes the everyday notion of size. Intuitively one can think of measure as length, weight, volume, distance from a point, probability, etc.

Consider a non-empty set \fs4 \Omega. Also consider a \fs4 \sigma-algebra on \fs4 \Omega, denoted with \fs4 \cal{F}. The pair \fs4 (\Omega, \cal{F}) is called a measurable space, since we can apply the measure function on the members of \fs4 \cal{F}. The members of \fs4 \cal{F} are called just measurable.

Contrary to intuition, not all subsets of \fs4 \Omega are (necessarily) measurable, and therefore there are subsets that don't belong in any \fs4 \sigma-algebra \fs4 \cal{F}. An important example of such subsets is the Vitali set.

A measure \fs4 \mu will be a function

Changed lines 12-13 from:
  • The empty set has measure zero
to:
  1. The empty set has measure zero
Changed lines 16-18 from:
  • The measure of a countable union of disjoint sets equals the sum of their measures. If \fs4 F_1, F_2, \ldots \in \cal{F} with \fs4 F_i \cap F_j = \emptyset for \fs4 i \neq j, then
\fs4 \mu(\cup_{i=1}^\infty F_i) = \sum_{i=1}^\infty \mu(F_i)
to:
  1. The measure of a countable union of disjoint sets equals the sum of their measures. If \fs4 F_1, F_2, \ldots \in \cal{F} with \fs4 F_i \cap F_j = \emptyset for \fs4 i \neq j, then
\fs4 \mu(\cup_{i=1}^\infty F_i) = \sum_{i=1}^\infty \mu(F_i)

We are particularly interested in measures that have the extra property that the total measure of \fs4 \Omega is one

\fs4 \mu(\Omega) = 1

If we look at the members of \fs4 \cal{F} as events then we can view this measure as the probability of these events, and therefore we call this measure a probability measure. Equipped with this probability measure, the measure space is called a probability space.

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November 03, 2006, at 12:50 AM by Kyriakos -
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  • The measure
to:
  • The measure of a countable union of disjoint sets equals the sum of their measures. If \fs4 F_1, F_2, \ldots \in \cal{F} with \fs4 F_i \cap F_j = \emptyset for \fs4 i \neq j, then
\fs4 \mu(\cup_{i=1}^\infty F_i) = \sum_{i=1}^\infty \mu(F_i)
Restore
November 02, 2006, at 02:25 PM by Kyriakos -
Added lines 1-10:

Consider a non-empty set \fs4 \Omega. Also consider a \fs4 \sigma-algebra on \fs4 \Omega, denoted with \fs4 \cal{F}. A measure \fs4 \mu will be a function

\fs4 \mu : \cal{F} \rightarrow [0, +\infty]

which satisfies the following properties:

  • The empty set has measure zero
\fs4 \mu(\emptyset) = 0
  • The measure
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