I like the book Probability, Statistics, and Random Processes for Electrical Engineering by Leon-Garcia (I used it to teach probability last semester). Of course, it is Electrical Engineering focused. It supresses the esoteric measure theory material.
The title of your book is funny because my first reaction is "Who but a mathematician would ever need measure theory anyway?" Of course I am being a bit snarky with that comment. I do think it is important for advanced students to at least know what measure theory is about. However, I feel it is more important for students to know the difference between a countably infinite set and an uncountably infinite set. Knowing the difference is essential for measure theory, but unfortunately some courses skip this, assume you already know it, and cover the less essential topics of sigma algebras.
In my humble opinion, you don't really need measure theory for probability or stochastic processes, although measure theory is certainly the foundation for those things. Similarly, Russell's Principia Mathematica is a foundation for basic arithmetic, but most people who use arithmetic (including mathematicians) have never read it (and certainly arithmetic existed before it). So, it is possible for you to learn and use something without going into detail on foundations. It is also good to know those foundations exist, especially if you eventually want answers to lingering questions.
The term "almost surely" is the most important one that you listed. It is synonymous with "with probability 1." For example, if you have a random variable $X$ that is uniformly distributed over the interval $[0,1]$, then $Pr[X \neq 1/2]=1$ and so almost surely $X$ is not $1/2$. Similarly, since the rational numbers in $[0,1]$ can be listed as $\{q_1, q_2, q_3, \ldots\}$ we have:
$$ Pr[\mbox{$X$ is rational}] = \sum_{i=1}^{\infty}Pr[X=q_i]=0 $$
and so $Pr[\mbox{$X$ is irrational}]=1$, which means $X$ is almost surely irrational.
Some sets are so complicated that they cannot have probabilities assigned to them. The term measurable describes a set that has a valid probability. A theorem that says "assuming the set is measurable" is just being precise, and you can ignore that phrase without worry. It just means they are restricting to the case where probabilities are defined (you cannot prove theorems otherwise). All the crazy theorems about measurability are designed to basically show that all practical sets of interest are measurable. In that sense, measure theory is self-defeating, since its most important results ensure you can safely ignore them.
Perhaps the most practical topics in measure theory are the convergence theorems, such as the Lebesgue dominated convergence theorem. These theorems tell you when you are allowed to pass a limit through an integral or through an expectation.
A "sigma algebra" is a class of sets defined so that all sets in the class can be measured. The "standard Borel sigma algebra" is a very large class of sets of real numbers. It is always possible to define the probability that a random variable falls in one of those sets. Since those sets are so extensive, every practical set you will ever work with will indeed be measurable (unless you end up working on foundational mathematical subjects or axiom-of-choice related set theory subjects).
Best Answer
Stochastic Calculus and Financial Applications by J. Michael Steele is the book for you, in my view. This is definitely an applied math book, but also rigorous. The author always keeps finance uses in mind although building concepts from the ground up. Some consider this "hard" but that's because they may not have the math training for it (some may never have done a proper proof, for instance). This will certainly not be a worry for you. Finally, the book is enjoyable and reveals the beauty of the subject.
I have also used Baxter and Rennie, but I found it a little painful. This book is written so that it is accessible to people who have done not more that 2 years of college math (it's enough if you know some calculus and other basics that you might have learned in high school or freshman year). That's great if you are in that boat, but it contorts itself to make that possible. If you have sone some measure theory even a while ago, you are no longer in that boat.
Here is another one by Shreve. Also great. But no one focuses solely on applications as sharply as Steele does without ever sacrificing rigor.
Then there are various specialized topics like stochastic volatility with good books, but that that's not a starting point. (Although, the first 2 chapters of this book are an interestingly different presentation of general stuff). You will also find a bunch of good books on option pricing. Course pdf on stochastic Calculus for finance and aplenty on google. Do look to see what you may like.
This book on Stochastic Calculus by Karatzas and Shreve is also great and many have gone to the industry with this as part of their training but perhaps leans too theoretical for your needs and is not specifically for finance.
My background in the area at that time was pretty much what you have now. Taught myself “streetsmart” probability when I decided on the career move and went from there. Please note, your probability questions in interviews (as opposed to the job) will usually not involve any measure-theoretic probability, just puzzles and problems that you can find in puzzle books and even here on MSE. But you are already in a job and know that.