6
$\begingroup$

Let $(X,A,\mu)$ be a measure space. Take a sequence $(f_n)_{n \in \mathbb N}$ of real-valued measurable bounded functions. Suppose $f_n \to f$ uniformly on $X$ and suppose that $\mu(X)<+\infty$. Then $ \int_X f_n d\mu \to \int_Xf d\mu $ when $n \to +\infty$.

This exercise is taken from Rudin, Real and Complex Analysis, chapter 1. I do not understand where I should use the hypothesis of boundedness of the functions. Indeed, $ \left \vert \int_X f-f_n d\mu \right\vert \le \int_X \vert f_n - f \vert d\mu \le \int_X \varepsilon d\mu = \varepsilon \mu(X) $ for $n$ sufficiently big.

Where do I use the boundedness of the functions? Do I need it in order to say $\vert \int_X f_n-f d\mu\vert \le \int \vert f-f_n \vert d\mu$?

2 Answers 2

7

Boundedness of the maps and finiteness of the measure space are used to be sure that $\int_Xf_nd\mu$ and $\int_Xfd\mu$ are real numbers.

  • 0
    Yes. Actually, it's a sufficient condition to assume boundedness. For example, integrability of all these functions is enough.2012-11-10
2

If the functions are not bounded, you may not be able to consider the absolute value of the difference. you might end up with infinity-infinity which is undefined.