Projection (measure theory)

Imagine you have a red ball and a blue ball. You want to put them into two different boxes - one for the red ball and one for the blue ball. One way to do this is by looking at each ball and checking its color before putting it in the correct box. This is called projection.

In math, projection is a way to separate different types of things. Instead of using colors, we use mathematical properties or attributes to separate things. For example, if we have a list of numbers, we can project them onto the even numbers and odd numbers. We put all the even numbers in one group and all the odd numbers in another group. This process of separating things based on their attributes is called projection.

In measure theory, projection is used to separate sets based on their size or measure. Measure is a concept used to assign a value to sets that tell us how big or small they are in a mathematical way. For example, the measure of a line segment is its length, and the measure of a circle is its area.

So, if we have a set of numbers, we can project it onto a smaller set based on its measure. For example, we can project all the numbers between 0 and 10 onto the interval between 0 and 5. This means that we only keep the numbers between 0 and 5 and throw away the rest. The result is a smaller set that contains only the numbers we want.

Overall, projection is a useful tool in mathematics to separate things based on their attributes or measure. It makes it easier to work with different sets by breaking them down into smaller, more manageable parts.