Understanding the definition of a measure on a sigma algebra

Key concept: This item is a definitions-and-examples prompt in measure theory. The best answer should restate the axioms clearly, explain the meaning of pairwise disjointness, and connect the definition to measure spaces.

Step by step

Core definition

A measure on a measurable space (X,\mathcal{A}) is a function

$$\mu:\mathcal{A}\to[0,\infty]$$

such that two conditions hold:

1. \mu(\varnothing)=0.
2. If A_1,A_2,\ldots are pairwise disjoint sets in \mathcal{A}, then

$$\mu\left(\bigcup_{i=1}^{\infty}A_i\right)=\sum_{i=1}^{\infty}\mu(A_i).$$

This second property is called countable additivity.

What pairwise disjoint means

The sets A_i are pairwise disjoint when

$$A_i\cap A_j=\varnothing \quad \text{for } i\ne j.$$

That condition matters because it prevents any overlap. If two sets overlap, then the same points would be counted twice, so the additivity formula would not apply in the same way.

Finitely additive versus countably additive

A function is finitely additive if the additivity rule holds only for finite disjoint unions. A measure is stronger: it must work for countably infinite disjoint families as well. That extra condition is one of the central ideas in measure theory and is what makes integration and limiting arguments possible.

Measure space language

The triple (X,\mathcal{A},\mu) is called a measure space. Here:

• X is the underlying set,
• \mathcal{A} is the sigma-algebra of measurable sets,
• \mu assigns a nonnegative size, length, area, probability, or mass to those sets.

Why this definition is important

The definition is designed so that measures behave well under limits and disjoint unions. It generalizes familiar notions such as length on intervals, area in the plane, and probability of events. Any function that meets the two axioms is immediately eligible to be used as a measure in later theorems.

Pitfall alert

A frequent mistake is to confuse a sigma-algebra with a measure. A sigma-algebra is a collection of sets closed under complements and countable unions; it does not assign sizes. Another common error is to think finite additivity is enough. In measure theory, countable additivity is essential, and many later results fail without it. It is also easy to forget that the codomain must be [0,\infty], not all real numbers, because measures are never allowed to be negative.

Try different conditions

If the definition were changed so that only finite disjoint unions were required, you would be talking about a finitely additive set function rather than a measure. For example, a variant question might ask: "Suppose \mu(\cup_{i=1}^n A_i)=\sum_{i=1}^n\mu(A_i) for disjoint sets. Is \mu automatically a measure?" The answer is no, because countable additivity is still missing. Another variant is the probability-measure version, where \mu(X)=1 is added as a third axiom. That extra condition turns a measure into a probability measure.

Further reading

sigma algebra, countable additivity, measure space