Measuring in Space

Measuring in Space

Measuring space is a basic object of the theory of measure. A branch of mathematics that studies generalized notions of volume and It contains an underlying set, the subset of this set that is feasible for measuring and the method that is used for measuring. The example of a measure space is a probability space and a measurable space consists of the first two components without a specific measure.

Example

X={1,2,3}

One possible σ algebra would be:

A₁={X,∅}

Then (X, A₁)is a measurable space. Another possible σ-algebra would be the power set on X:

A₂=P(X)

With this, a second measurable space on the set X is given by (X, A₂)

The most important classes of measure spaces are defined by the properties of their associated measures. This includes

• • Probability space is a measure space where the measure is a probability measure.
  • Finite measure spaces, where the measure is a finite measure
  • σ finite measure spaces, where the measure is a -finite measure

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