Intuitively for me probability theory is a bit clunky before measure theory. Like we have a different equation for expectations if something is continuous vs a discrete distribution. We use probability density functions (pdf) vs a probability mass function (pmf) and etc.
We know this stuff is basically getting at the same underlying quantities. Now imagine a distribution over both continuous and discrete. For example something that measures temperature but breaks after a certain threshold. What does the expectation be for such an instrument? Imagine a distribution on different sized arrays of real numbers. How do you define a valid density function? Measure theory gives you the formalisms for those kinds of problems. You in practice don't need it very often but it keeps you on firm ground when you do.
We know this stuff is basically getting at the same underlying quantities. Now imagine a distribution over both continuous and discrete. For example something that measures temperature but breaks after a certain threshold. What does the expectation be for such an instrument? Imagine a distribution on different sized arrays of real numbers. How do you define a valid density function? Measure theory gives you the formalisms for those kinds of problems. You in practice don't need it very often but it keeps you on firm ground when you do.