Lebesgue measure

In measure theory, a part of mathematics, the Lebesgue measure is a way to give a size to different parts of space. It is named after the French mathematician Henri Lebesgue. For simple shapes in one, two, or three dimensions, this measure is the same as regular length, area, or volume. But it works for more complicated shapes too.

We often call this size the n-dimensional volume, n-volume, hypervolume, or just volume. It is very important in a part of math called real analysis, especially for a special way of adding up values called Lebesgue integration.

Henri Lebesgue first described this idea in 1901, and he wrote about it along with his new way of integration in his dissertation called Intégrale, Longueur, Aire in 1902. Sets that we can give a Lebesgue measure to are called Lebesgue-measurable. We write the measure of one of these sets as λ ( A ).

Definition

The Lebesgue measure is a way to find the size of different shapes in space. It works for lines, flat shapes, and even more complex objects.

For simple shapes like straight lines or flat areas, the Lebesgue measure is just the usual length or area you learn about in school. But for more complicated shapes, mathematicians use a special method to figure out their size. They cover the shape with lots of tiny intervals or boxes, add up the sizes of those boxes, and then make that total as small as possible. This gives the outer size of the shape.

Some shapes fit this method perfectly, and for these, the Lebesgue measure matches the outer size. These special shapes are called Lebesgue-measurable. But there are also strange shapes where this method doesn’t work quite right — these are called non-measurable sets.

Examples

In mathematics, the Lebesgue measure is a way to find the size of different shapes or sets of numbers. For simple shapes like lines or flat areas, the Lebesgue measure matches what we usually think of as length or area. For example, a closed line from point a to point b has a length of b − a.

For two-dimensional shapes, like rectangles made by multiplying two intervals, the Lebesgue measure gives the area by multiplying the lengths of the sides. Some special sets, like the Cantor set or very thin curves, have a Lebesgue measure of zero, meaning they are considered to have no size in this way of measuring.

Properties

The Lebesgue measure helps us understand sizes of different shapes in space. It works for one, two, and three dimensions, matching what we know about length, area, and volume.

For example, if you have a rectangle made by multiplying lengths together, the Lebesgue measure gives the correct area. It also works well with many different shapes and combinations of shapes, always giving a non-negative size. This makes it a useful tool in mathematics for measuring all sorts of sets.

Null sets

Main article: Null set

A null set is a special kind of group of points in space. It is so small that, no matter how tiny a piece of space you pick, you can cover all the points in the null set using just a few long, thin slices. These slices can be made so small that their total size is almost nothing.

Some groups of points that are easy to list, like all the whole numbers, are also null sets. Even if a group of points looks simple, it might still take up space in a special way. For example, there are special groups of points that seem simple but still take up some amount of space.

To check if a group of points follows the rules of Lebesgue measure, we often find a simpler group that is almost the same, except for a null set. This helps us understand how these points behave in space.

Construction of the Lebesgue measure

The Lebesgue measure is a way to find the size of different shapes in space. It works for lines, flat shapes, and even more complex forms. To build this measure, we start with simple boxes. A box in space is made by picking a starting point and an ending point in each direction.

We can find the size of these boxes by multiplying how far they stretch in each direction. For any shape, we can cover it with many small boxes and add up their sizes. The Lebesgue measure of a shape is the smallest total size we can get this way.

Some very special shapes might not have a clear size using this method, but these are rare and need extra ideas from set theory to understand.

Relation to other measures

The Borel measure works the same as the Lebesgue measure for certain sets, but there are many more sets that the Lebesgue measure can describe. The Lebesgue measure on Rⁿ is a special type of Borel measure, but not all such measures are Lebesgue measures. The Borel measure stays the same when you move it around, but it isn't "complete".

The Haar measure can be used with any locally compact group and extends the ideas of the Lebesgue measure. The Hausdorff measure is another extension of the Lebesgue measure, useful for measuring lower-dimensional sets like surfaces or curves in R³, and also for measuring fractal sets. Unlike Hausdorff measure, Hausdorff dimension is a different concept.

It is also known that there is no version of the Lebesgue measure that works in infinite-dimensional spaces.