measure (mathematics)

We can measure the length of a curve, the area of a surface, the volume of some object. From a mathematical perspective, the curve, surface and object are just sets of points in space, so length, area and volume give us a way to measure the extent of one, two and three-dimensional sets respectively. This is the fundamental idea of a measure: a rule that associates to sets a number that quantifies the size of the set. Such a concept was developed to measure the length, area, or volume of things that don't have a naturally obvious length, area or volume.

The following are intuitive explanations of the properties of a measure:

1. Non-negativity: Length, area and volume can in principle be any positive value; we just need to find a long enough string, a big enough sheet of paper, or a large enough cube. They can be zero too, a point is a curve with zero length, a line is rectangle with zero width and a plane is a solid with no depth. If we assume that the universe has infinite space, we can extend this argument to infinity.
   

2. Additivity: We can measure the length of a piece of rope. If we cut this rope into three non-overlapping pieces, then the lengths of these three pieces must combine to be the same as the length of the original rope. This same fact is true no matter how we partition any region. We call this property additivity. Similarly, to divide a set \[\mathcal{A}\], e.g. the points on a curve, rectangular region or object, into non-overlapping pieces is to find a collection of pairwise disjoin sets \[\mathcal{A}_{1},\mathcal{A}_{2},\dots\] such that \[\mathcal{A}=\underset{i}{\cup}\,\mathcal{A}_{i}\].
   

   Additionally, any measure of size for which the additivity property holds must have another important property, monotonicity. Using our previous example, the length of a segment of a rope can not be more longer than the entire rope. More generally, if \[\mathcal{A}\subset \mathcal{B}\], then the size of \[\mathcal{A}\] must be no greater than the size of \[\mathcal{B}\].
   

3. Empty set: Since the magnitude of length, area and volume derives from the extent of the curve, surface, or object, and since the empty set by definition has no points and thus no extent, it makes sense to take the length, area and volume of the empty set as zero.
   

Note that the measure is just a function with a strange domain, and is usually denoted by \[m\] or \[\mu\].

Let \[\mathcal{X}\] be any set. A measure on \[\mathcal{X}\] is a function \[\mu\] that maps the set of subsets of \[\mathcal{X}\] to \[\left[ 0,\infty \right]\] that satisfies:

• \[\mu \left( \emptyset \right)=0\]
  
• The countable additivity property, i.e. for any countable and pairwise disjoint collection of subsets of \[\mathcal{X}\], \[\mathcal{A}_{1},\mathcal{A}_{2},\dots\], \[\mu \left( \bigcup_{a}\,\mathcal{A}_{i} \right)=\sum_{i}\mu \left( \mathcal{A}_{i} \right)\]
  
• If \[\mathcal{A}\subset \mathcal{X}\], \[\mu \left( \mathcal{A} \right)\ge0\]
  

A simple measure function is a counting measure. Assume \[\mathcal{X}\] is countable, e.g. \[\mathcal{X}=\mathbb{Z}\]. Then a natural measure is the counting measure, mathematically written as \[\mu \left( A \right)\], which simply counts the number of points in a subset \[A\], i.e. \[\mu(\left\{ 0,1 \right\})=2\] and \[\mu \left( \left\{ 2,4,6,8,\dots \right\} \right)=\infty\].