The Unapologetic Mathematician

Mathematics for the interested outsider

Properties of Integrable Functions

Now we’ve got a definition of the integral of a wider variety of functions than before, so let’s look over the basic properties.

First of all, from what we know about convergence in measure and algebraic and order properties of integrals of simple functions, we can see that if f and g are integrable functions and \alpha is a real number, then so are the absolute value \lvert f\rvert, the scalar multiple \alpha f, and the sum f+g. As special cases, we can see that the positive and negative parts

\displaystyle\begin{aligned}f^+&=\frac{1}{2}(\lvert f\rvert+f)\\f^-&=\frac{1}{2}(\lvert f\rvert-f)\end{aligned}

are both integrable.

If E is a measurable set and \{f_n\} is a mean Cauchy sequence of integrable simple functions converging in measure to f, then it should be clear that \{f\chi_E\} is mean Cauchy and converges in measure to f\chi_E. Thus we can define

\displaystyle\int\limits_Ef\,d\mu=\int f\chi_E\,d\mu

Since integrals of simple functions are linear, and limits are linear, we immediately conclude that

\displaystyle\int\alpha f+\beta g\,d\mu=\alpha\int f\,d\mu+\beta\int g\,d\mu

as before, but now for general integrable functions. Similarly, if f is nonnegative a.e., we can find a sequence of nonnegative simple functions converging a.e. (and thus in measure) to f. The integral of each of these functions is nonnegative, and so their limit must be as well. That is, if f\geq0 a.e., then

\displaystyle\int f\,d\mu\geq0

Now, all our properties that we proved using only these two linearity and order properties follow. If f\geq g a.e., then

\displaystyle\int f\,d\mu\geq\int g\,d\mu

For any two integrable functions f amd g we find

\displaystyle\int\lvert f+g\rvert\,d\mu\leq\int\lvert f\rvert\,d\mu+\int\lvert g\rvert\,d\mu

and

\displaystyle\left\lvert\int f\,d\mu\right\rvert\leq\int\lvert f\rvert\,d\mu

If \alpha and \beta are real numbers so that \alpha\leq f(x)\leq\beta for almost all x\in E, then we have

\displaystyle\alpha\mu(E)\leq\int\limits_Ef\,d\mu\leq\beta\mu(E)

and if an integrable function f is a.e. nonnegative, then its indefinite integral is monotone.

It takes a bit of work, though, to check that the indefinite integral \nu of an integrable function f is absolutely continuous. If \{f_n\} is a mean Cauchy sequence converging in measure to f, then we have

\displaystyle\left\lvert\int\limits_Ef\,d\mu\right\rvert\leq\left\lvert\int\limits_Ef_n\,d\mu\right\rvert+\left\lvert\int\limits_Ef\,d\mu-\int\limits_Ef_n\,d\mu\right\rvert

We know that the indefinite integrals \nu_n of the f_n are uniformly absolutely continuous, so we have control over the size of the first term on the right. The second term on the right can be kept small by choosing a large enough n, since \{f_n\} converges in measure to f.

Finally, the indefinite integral \nu of f is countably additive; since \{f_n\} is a mean Cauchy sequence of simple functions we can write \nu as the limit \nu(E)=\lim_n\nu_n(E), and we know that this limit is countably additive.

June 3, 2010 Posted by | Analysis, Measure Theory | 3 Comments

   

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