Lebesgue's lemma
Lebesgue's lemma is a fancy math rule that helps us understand how smaller pieces can add up to make something bigger. Imagine you have a big cake and you want to cut it into small pieces, but you don't want to lose or waste any cake. Lebesgue's lemma tells us that if we cut the cake into smaller pieces, then the total amount of cake we have will always be the same, no matter how small we cut the pieces. That means we can cut the cake into as many small pieces as we want, and the total amount of cake we have will stay the same.
Let's take another example. Imagine we have a giant pool filled with water, and we want to measure how much water is in the pool. But the pool is so big that we can't measure it all at once. So we use Lebesgue's lemma to help us. We can divide the pool into smaller sections, like squares or rectangles, and measure the amount of water in each section. Then we add up all the measurements from each section to get the total amount of water in the pool. Lebesgue's lemma tells us that no matter how small or big the sections are, the total amount of water in the pool will always be the same.
So, in general, Lebesgue's lemma is a handy rule to have when dealing with a big complicated thing but want to divide it into smaller, simpler parts for understanding or analysis. It tells us that no matter how small or big we divide the thing, the whole will remain the same, and we won't lose anything.
Let's take another example. Imagine we have a giant pool filled with water, and we want to measure how much water is in the pool. But the pool is so big that we can't measure it all at once. So we use Lebesgue's lemma to help us. We can divide the pool into smaller sections, like squares or rectangles, and measure the amount of water in each section. Then we add up all the measurements from each section to get the total amount of water in the pool. Lebesgue's lemma tells us that no matter how small or big the sections are, the total amount of water in the pool will always be the same.
So, in general, Lebesgue's lemma is a handy rule to have when dealing with a big complicated thing but want to divide it into smaller, simpler parts for understanding or analysis. It tells us that no matter how small or big we divide the thing, the whole will remain the same, and we won't lose anything.