Measure (mathematics) Facts for Kids

Informally, a measure has the property of being monotone in the sense that if

A

is a subset of

B,

the measure of

A

is less than or equal to the measure of

B.

Furthermore, the measure of the empty set is required to be 0. A simple example is a volume (how big an object occupies a space) as a measure.

In mathematics, a measure is a way to give a "size" to parts of a set. Think of it like measuring length, area, or volume in geometry. It's also used for things like mass or the probability of an event happening.

These different ideas of "size" actually have a lot in common. Mathematicians created the idea of a "measure" to study them all together. Measures are super important in probability theory (which deals with chance) and integration theory (which is about adding up tiny pieces).

The basic idea of measuring things goes back to ancient Greece. For example, Archimedes tried to figure out the area of a circle. But the modern mathematical theory of measures only really started in the late 1800s and early 1900s. Many smart mathematicians like Henri Lebesgue helped build this field.

What is a Measure?
Examples of Measures
Basic Rules of Measures
- Monotonicity
- Measure of Combined Sets
Other Important Ideas
- Completeness
- Sigma-finite Measures
Non-measurable Sets
Generalizations
See also

What is a Measure?

Countable additivity of a measure

\mu

: The measure of a countable disjoint union is the same as the sum of all measures of each subset.

Imagine you have a big set of things, let's call it $X$ . A measure is like a special rule that tells you the "size" of certain parts (or subsets) of $X$ . These parts must be "measurable sets," which means they are part of a special collection called a $\sigma$ -algebra.

For something to be called a "measure," it must follow these rules:

No Negative Sizes: The "size" (measure) of any part must be zero or a positive number. You can't have a negative length or area!
Empty Set is Zero: The measure of an empty set (a set with nothing in it) is always 0.
Adding Up Pieces: If you have a collection of parts that don't overlap (they are disjoint sets) and you combine them, the measure of the combined part is the sum of the measures of each individual part. This is called countable additivity. For example, if you have two separate pieces of land, the total area is the sum of their individual areas.

When you have a set $X$ and a collection of its measurable parts, it's called a measurable space. If you add a measure to it, it becomes a measure space.

A special type of measure is a probability measure. This is a measure where the "size" of the entire set $X$ is exactly 1. This is useful for probabilities, where the chance of something definitely happening is 1 (or 100%).

Examples of Measures

There are many different types of measures used in mathematics and science:

Counting Measure: This is the simplest measure! It just counts how many items are in a set. For example, the counting measure of the set {apple, banana, orange} is 3.
Lebesgue Measure: This is the most common measure in everyday life. It's used to find the length of lines, the area of shapes, and the volume of objects in space. For example, the Lebesgue measure of a line segment from 0 to 1 is 1.
Probability Measure: As mentioned, this measure tells you the chance of an event happening. The total probability of all possible outcomes is always 1. For example, the probability measure of flipping a coin and getting heads is 0.5.
Dirac Measure: This measure is special because it only "sees" if a specific point is in a set. If the set contains that point, its measure is 1; otherwise, it's 0. Imagine a tiny sensor that only detects if a specific grain of sand is present.
Haar Measure: This is a more advanced measure, but it's like a generalized Lebesgue measure for certain mathematical structures called "topological groups." It helps measure "size" in a way that stays the same even if you move or rotate things.
Hausdorff Measure: This measure is used for more complex shapes, like fractals, which can have "dimensions" that aren't whole numbers (like 1.5 or 2.3).

In physics, measures are used to describe things like the distribution of mass in space. For example, how much mass is in a certain region.

Basic Rules of Measures

Let's look at some basic rules that all measures follow:

Monotonicity

If you have two measurable sets, $E_1$ and $E_2$ , and $E_1$ is completely inside $E_2$ (meaning $E_1 \subseteq E_2$ ), then the measure of $E_1$ will always be less than or equal to the measure of $E_2$ . This makes sense: a smaller piece can't have a larger size than a bigger piece that contains it!

Measure of Combined Sets

Countable Subadditivity

If you have a collection of measurable sets $E_1, E_2, E_3, \ldots$ (they don't have to be separate), the measure of their combined total (their union) will be less than or equal to the sum of their individual measures. It's like saying if you overlap some pieces of paper, the total area they cover is less than or equal to the sum of their individual areas.

Continuity from Below

Imagine you have a sequence of measurable sets that are growing bigger and bigger, with each set containing the previous one ( $E_1 \subseteq E_2 \subseteq E_3 \subseteq \ldots$ ). As these sets grow, their measures get closer and closer to the measure of their final, combined set (their union).

Continuity from Above

Now imagine you have a sequence of measurable sets that are shrinking, with each set being contained within the previous one ( $E_1 \supseteq E_2 \supseteq E_3 \supseteq \ldots$ ). If at least one of these sets has a finite (not infinite) measure, then as these sets shrink, their measures get closer and closer to the measure of their final, overlapping part (their intersection).

Other Important Ideas

Completeness

A measurable set $X$ is called a null set if its measure is 0. Think of a single point on a line; its length (Lebesgue measure) is 0, so it's a null set. A part of a null set is called a negligible set.

A measure is called complete if every negligible set is also measurable. This means if a set is "small enough" to be part of something with zero measure, then you can also measure it.

Sigma-finite Measures

A measure space is called finite if the measure of the entire space $X$ is a finite number (not infinity).

A measure is called σ-finite (pronounced "sigma-finite") if you can break down the entire set $X$ into a countable number of measurable parts, where each part has a finite measure.

For example, the set of all real numbers with the standard Lebesgue measure (length) is σ-finite. You can cover the entire real line with countable intervals like [0,1], [1,2], [2,3], and so on, each having a finite measure (length 1).

Non-measurable Sets

It might seem strange, but not every subset of space can be "measured" in a way that follows all the rules of a measure. If we assume a certain mathematical rule called the axiom of choice is true, then it's possible to show that some sets simply don't have a well-defined Lebesgue measure. These are called non-measurable sets. They are very complex and hard to imagine!

Generalizations

Sometimes, mathematicians use "measures" that can take on different kinds of values:

Signed Measure: This is like a measure, but its values can be negative. Think of electrical charge, which can be positive or negative.
Complex Measure: This is a measure whose values can be complex numbers (numbers that include an imaginary part).
Finitely Additive Measure: This is an older idea, where the "adding up pieces" rule only applies to a finite number of parts, not an infinite (countable) number.

Measure (mathematics) facts for kids

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