# Thread: Exercise in Real Analysis

1. ## Exercise in Real Analysis

It's the Ex.24 in Chapter 3 Real Analysis (Stein) Does anyone know how to solve it?

Suppose F is an increasing function on [a,b]
a). Prove that we can write F= $\displaystyle F_A +F_C +F_J$
where $\displaystyle F_A$ is absolutely continuous; $\displaystyle F_C$ is continuous, but $\displaystyle F'_C = 0$ for a.e. x;
$\displaystyle F_J$ is a jump function;
b). Moreover, each component is uniquely determined up to an additive constant.

2. Originally Posted by Gintoki
It's the Ex.24 in Chapter 3 Real Analysis (Stein) Does anyone know how to solve it?

Suppose F is an increasing function on [a,b]
a). Prove that we can write F= $\displaystyle F_A +F_C +F_J$
where $\displaystyle F_A$ is absolutely continuous; $\displaystyle F_C$ is continuous, but $\displaystyle F'_C = 0$ for a.e. x;
$\displaystyle F_J$ is a jump function;
b). Moreover, each component is uniquely determined up to an additive constant.
What does "a.e" mean?

3. "A.e." is a technical term meaning "almost everywhere". It means that the property in question holds everywhere except on a set of measure zero.

4. Originally Posted by Ackbeet
"A.e." is a technical term meaning "almost everywhere". It means that the property in question holds everywhere except on a set of measure zero.
Well I know what "almost everywhere" means, but I don't know what "for a.e." means. I don't even know what "lol" means.

5. "For almost every x"