But no need to get all worked up! We’re here to help you understand it in a way that won’t put you to sleep (or worse, bore you to tears).

To set the stage: what exactly do we mean by convergence of measurable functions? Well, let’s say we have two functions f and g defined on some measure space X. If for every ε > 0 there exists a set A such that m(X A) < ε (where m is the measure on X), then we can say that f converges to g almost everywhere with respect to m, or in other words, f converges to g in measure. Now, you might be thinking: "That's all well and good, but what does this actually mean?" Well, let me give you an example. Let X = [0,1], equipped with the Lebesgue measure (which is just a fancy way of saying that we're measuring length). Define f(x) to be x^2 for 0 <= x < 1/2 and f(x) to be 1 x^2 for 1/2 <= x <= 1. This function has a jump discontinuity at x = 1/2, which means that it's not continuous everywhere on X (but we can still integrate it!). Now let g(x) be the same as f(x), except with the value of f(1/2) changed to 0.5. This function is also discontinuous at x = 1/2, but in a different way: instead of having a jump discontinuity, it has a removable singularity (which means that we can make it continuous by changing the value of f(1/2) slightly). So what happens when we try to converge these two functions? Well, let's say we want to find out if g(x) converges to f(x) in measure. To do this, we need to show that for every ε > 0 there exists a set A such that m(X A) < ε and such that the difference between g(x) and f(x) is small on A (i.e., |g(x) f(x)| < ε). In this case, we can take A to be [0,1/2] U (1/2 ε, 1/2 + ε], where the second interval is included to account for any points near x = 1/2 that might cause problems. This set has measure less than or equal to 3ε (since it's a union of two intervals with total length less than or equal to 4ε), and on this set we have |g(x) f(x)| < ε, since g(x) = f(x) for x <= 1/2 and the difference between them is at most 0.5 (which is less than ε if ε > 0.5).

It might not be as exciting as watching paint dry, but it’s definitely more interesting than listening to someone drone on about the intricacies of Lebesgue integration (which is what we were doing before this).