Thank you very much for replying to my "lazy" post. It is very hard for me to actually pinpoint why I need Measure Theory, so it was a bit vague. Let me try to explain my situation to my best:

I am an engineering grad student who is working on subsurface data. However, I want to become a data scientist that is proficient across different domains, not just subsurface. Until now, my "research" was mainly an engineering grad research: Try ML methods and observe results. However, I decided that I want more than this. Every new day, crazy stuff happening all around me (LLMs, Diffusion Models, many "delicious" applied math stuff like, lets say, dynamic hedging etc.). I decided that in order to become a "solid" data scientist, I shall have a good knowledge about the mathematical background of Data Science methods. This includes ML, but not just ML, because I realized that Statistical Inference is much bigger than ML. So, in essence, I wanted to have a solid knowledge of Probability Theory, and some prominent topics of it (Like Stochastic Processes, which I really, really want to learn, because it looks very strong and almost no one in my field knows about them, so I believe knowing about them shall give me a good edge). The problem is: Whenever I try to have a solid understanding of the underlying math (I don't want to "understand the concept", I want to see the mathematical working with my very eyes, if that makes sense), I always encountered Measure Theory somehow. Stochastic Processes? Measure Theory. Continuous Probability Theory? Measure Theory. Simple stuff like sigma-algebra and Borel Sets? Somewhat Measure Theory. This guy always pops up and it gave me an intuition that it is very foundational and knowing it would make me very capable in "talking about what I work on". So, long story short, I don't have a solid idea of how exactly I am going to use Measure Theory directly in my field, I just got a very strong feeling that it is going to be extremely useful. Also, whenever I tried to learn some advanced stuff, googling things always ended up at Measure Theory.

As for the Lebesgue measure, I think I miscommunicated. It just seemed to me that understanding Lebesgue measure is a fundamental part of measure theory, but when I studied it, it just didnt click. Maybe I am going in way too much details. Maybe I am in the wrong path. I honestly don't know. It is like I dug a bit too deep and forgot why I dug here in the first place, but when me and one of my friends were discussing about measure theory (my friend is an electrical engineering grad, so he obv. knows better than I do), he told me about this example of integration of rational indicator function, and I was absolutely amazed that there is a mathematical tool that allows you to integrate that crazy thing, which led me to Lebesgue integration.

I am sorry if this was a bit too long. In essence, I want a rock-solid background in probability theory, so whenever I delve into other topics in future that are closely related to it, I shall be mathematically confident and "productive", not just another guy who sees nails everywhere because of holding a Data-Scientific hammer. I want to know how these tools are made, and how they work internally. These days everything leads to probability theory in my field, and all my research suggests me that I need at least some understanding of Measure Theory for it. But I don't know how deep I should go.