Lebesgue's decomposition theorem

Lebesgue's decomposition theorem is like taking a toy box and separating all the toys into different groups so that they don’t mix up or create a mess.

In mathematics, it means separating a function into different parts, so we can understand it better.

Let's pretend that we have a function that tells us how much water is in a glass at different times of the day. We want to understand the function, but it's a bit tricky because the water level is different at different times of the day.

Lebesgue's decomposition theorem says we can break down this function into two parts: the increasing part and the decreasing part.

The increasing part is like the water level going up, and we can understand it better by looking at how it's going up over time. The decreasing part is like the water level going down, and we can understand it better by looking at how it's going down over time.

This way, we understand the function better, and it's like separating toys into different groups, so they don't make a mess.