00:01
So in this problem, we're going to try to prove the proposition that if f is measurable, then f truncated at x, which is defined as a for values of x where f is bigger than a and f of x for values of x where f is less than or equal to a is also going to be measurable.
00:20
So first we suppose that f is measurable.
00:23
Suppose f is measurable.
00:29
And then after that, we're going to use a very, very useful fact, and that is that g, a function g, is measurable.
00:42
Measurable if and only if for all b, for all real numbers b, the pre -image of g, the pre -image of the interval negative infinity to b, close bracket, under g is measurable.
01:06
Okay, so this, in fact, any sort of, you can replace this interval with any sort of interval that involves an infinity and b, and this theorem that says all of these facts are true, that all of these statements are equivalent, should be somewhere, should have been learned by this point, and you should cite it properly if you're using a textbook.
01:32
Okay, so now let's use this fact to prove that f is, f truncated at a, is measurable.
01:40
So if b, right, so if b, so okay, sorry, consider any b, and we want to prove that we want to show that the pre -image of negative infinity to b under the f truncated at a is measurable.
02:11
Okay, and so the easy case, the easier case, is that suppose b is bigger than a...