Measurable Functions

# Measurable Functions

So far we have looked at three major classes of functions.

On the Step Functions on General Intervals page we said that a function $f$ on a general interval $I$ is a step function if there exists a closed and bounded interval $[a, b] \subseteq I$ such that there exists a partition $P = \{ a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$ where $f$ is constant on each open subinterval $(x_{k-1}, x_k)$ (for all $k \in \{ 1, 2, ..., n \}$) and such that $f(x) = 0$ for all $x \in I \setminus [a, b]$. The set of all step functions on $I$ was denoted $S(I)$.

On the Upper Functions and Integrals of Upper Functions page we said that a function $f$ on a general interval $I$ is an upper function if there exists an sequence of step functions, call it $(f_n(x))_{n=1}^{\infty}$ that is increasing and converges to $f$ almost everywhere on $I$ and such that the numerical sequence $\displaystyle{�406�\lim_{n \to \infty} \int_I f_n(x) \: dx}$ is finite. The set of all upper functions on $I$ was denoted $U(I)$.

On the The Lebesgue Integral page we said that a function $f$ is a Lebesgue integrable function on $I$ if there exists upper functions $u(x)$ and $v(x)$ on $I$ such that $f(x) = u(x) - v(x)$. The set of all Lebesgue integrable function on $I$ was denoted $L(I)$.

We will now look at an every broader class of functions called measurable functions which we define below.

 Definition: A function $f$ is said to be a Measurable Function on an interval $I$ if there exists a sequence of step functions, $(f_n(x))_{n=1}^{\infty}$ on $I$ that converges to $f$ almost everywhere on $I$. The set of all measurable functions on $I$ is denoted $M(I)$.

Consider a function $f$ which is Lebesgue integrable on $I$. Then $f$ is the difference of two upper functions on $I$, say $u$ and $v$, where $f = u - v$. But each upper function is the limit of a convergent generating sequence of step functions $(u_n(x))_{n=1}^{\infty}$ and $(v_n(x))_{n=1}^{\infty}$.

If we let $(w_n(x)) = (u_n(x) - v_n(x))_{n=1}^{\infty}$ then the sequence $(w_n(x))_{N=1}^{\infty}$ is a sequence of step functions that converges to $u - v = f$. So, every Lebesgue integrable function is a measurable function. Thus the following inclusion holds for any interval $I$:

(1)
\begin{align} \quad S(I) \subset U(I) \subset L(I) \subset M(I) \end{align}

One may think that the set of Lebesgue integrable functions equals the set of Measurable functions. This is NOT the case, however, giving an example of a measurable function that is not Lebesgue integrable is difficult.