This Springer Undergraduate Series book is a great intro to measure theoretic probability, it's very suitable for self-study and has hints/partial solutions to many of the exercises. It is a very clearly written book. It also has some brief mathematical finance applications (2-3 pages) at the end of most chapters.
It's an ideal intro text imo for someone either with absolutely no prior knowledge or very little knowledge of measure theory. Very easy-going. More exercises would help though.