Unlock the Secrets of Measurable Functions: 6 Surprising Consequences of Proposition 34 Revealed!

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**Proposition 34**.: _Suppose \(f\) and \(g\) are measurable functions, and suppose the \(c\in\mathbb{R}\). Then the functions_

\[cf,\ f+g,\ fg,\ |f|,\ f\wedge g,\quad\text{and}\quad f\lor g\quad\text{are measurable}.\]

Proof

: We leave it to the reader to prove \(cf\) and \(fg\) are measurable. For the sum \(f+g\) note that if \((f+g)(x)>a\) then there is a rational number \(r\) such that \(f(x)>r>a-g(x)\), therefore

\[\{f+g>a\}=\bigcup_{r\in\mathbb{Q}}\left(\{f>r\}\cap\{g>a-r\}\right)\]

which is a countable union of measurable sets. For \(|f|\) write

\[\{|f|>a\}=\{f>a\}\cup\{f<-a\}\]

and for \(f\wedge g\) and \(f\lor g\) we also have:

\[\{f\wedge g>a\}=\{f>a\}\cap\{g>a\},\]

\[\{f\lor g>a\}=\{f>a\}\cup\{g>a\}.\qed\]

Now we turn our attention to a collection of measurable functions sharing a common domain \(D\) taking values in the extended real line. We are interested in a sequence of measurable functions \(f_{n}:D\to[-\infty,\infty]\).

Proof

: Let \(a\in\mathbb{R}\); then when we say \(\sup_{n}f_{n}>a\) we mean \(f_{n}(x)>a\) for some \(n\), and conversely

\[\{\sup f_{n}>a\}=\bigcup_{n=1}^{\infty}\{f_{n}>a\},\]

which is measurable because each \(f_{n}\) is measurable. Since

\[\inf_{n}f_{n}=-\sup_{n}(-f_{n}),\]

inf’s are measurable since sup’s are measurable.

**Corollary 22**.: _Let \((f_{n})\) be a sequence of measurable functions. Then both \(\limsup_{n\to\infty}f_{n}\) and \(\liminf_{n\to\infty}f_{n}\) are measurable._

Proof

: This directly follows from

\[\limsup_{n\to\infty}f_{n}=\inf_{n}\left(\sup_{k\geq n}f_{k}\right)\quad\text{ and}\quad\liminf_{n\to\infty}f_{n}=\sup_{n}\left(\inf_{k\geq n}f_{k}\right).\qed\]

Now we are ready for a major theorem which guarantees that the collection of measurable functions is closed under pointwise limits. This tells us that the class of measurable functions are actually quite large.

**Theorem 111**.: _If \((f_{n})\) is a sequence of measurable functions and if_

\[f(x)=\lim_{n\to\infty}f_{n}(x)\]

_exists for all \(x\in D\), then the limit function \(f\) is measurable._

Proof

: Since

\[f(x)=\limsup_{n\to\infty}f_{n}(x)=\liminf_{n\to\infty}f_{n}(x),\]

this is a consequence of the above corollary.

**Remark 61**.: Suppose \(f\) is a measurable function. Then we divide \(f\) into its positive and negative parts as follows:

\[f^{+}(x)=\sup(f(x),0)\qquad f^{-}(x)=\sup(-f(x),0)=-\inf(f(x),0).\]

The positive and negative parts of the function are as illustrated in Figure 2.22:

Figure 2.22: Positive and negative parts of a function

Observe that both \(f^{+}\) and \(f^{-}\) are non-negative, measurable and \(|f|=f^{+}+f^{-}\); furthermore \(f=f^{+}-f^{-}\) provided that this expression makes sense.

**Definition 93** (Almost everywhere).: A statement is said to hold _almost everywhere_ (a.e.) if it is true except on a set of measure zero.

For example, the characteristic function of the set of rationals defined as:

\[\chi_{\mathbb{Q}}(x)=\left\{\begin{array}{ll}1&\text{if}\;\;x\in\mathbb{Q} \\ 0&\text{if}\;\;x\notin\mathbb{Q}\end{array}\right..\]

We have \(\chi_{\mathbb{Q}}=0\) a.e. because it is only nonzero on rationals and the rational numbers being a countable set have measure zero. We shall say

\[f(x)=g(x)\;\;\text{a.e. on a set $\mathrm{D}$}\quad\text{if}\quad m(\{x:f(x) \neq g(x)\})=0.\]

It is not hard to show that if \((f_{n})\) is a collection of measurable functions and

\[\lim_{n\to\infty}f_{n}(x)=f(x)\quad\text{a.e.},\]

then \(f\) is measurable, i.e., the collection of measurable functions is closed even under pointwise a.e. limits.

**Definition 94**.: A _simple function_\(\varphi\) is a finite sum

\[\varphi=\sum_{k=1}^{N}a_{k}\chi_{E_{k}},\]

where \(\chi_{E_{k}}\) is the characteristic function of \(E_{k}\) defined earlier, each \(E_{k}\) is a measurable set of finite measure and the \(a_{k}\) are constants. The _canonical form_ of \(\varphi\) is the unique expression as in above where the numbers \(a_{k}\) are distinct and nonzero.

**Example 92**.: Let

\[\varphi(x)=\begin{cases}0&\text{if}\;\;x<0\;\text{or}\;\;x>2\\ 1&\text{if}\;\;0\leq x\leq 1\\ 2&\text{if}\;\;1

then we can write \(\varphi(x)=\chi_{[0,1]}+2\chi_{(1,2]}+0\chi_{(-\infty,0)\cup(2,\infty)}\) which is certainly not in canonical form.

When we define Lebesgue integration we will need the canonical form of \(\varphi\). However, for the following theorem on the approximation of measurable functions a general definition of simple functions is enough. The structure of measurable functions provides an insight about their relation to simple functions. The following theorem is about approximating non-negative measurable functions by simple functions.

**Theorem 112** (Approximation by simple functions).: _If \(f:D\to[0,\infty]\) is a non-negative measurable function, then we can find an increasing sequence of non-negative simple functions \((\varphi_{n})\) with_

\[0\leq\varphi_{1}\leq\varphi_{2}\leq\cdots\leq f\]

_such that \((\varphi_{n})\) converges pointwise to \(f\) everywhere on \(D\), and that \((\varphi_{n})\) converges uniformly to \(f\) on any set where \(f\) is bounded._

Proof

: The general idea of the proof is as follows (readers are advised to fill in the details). Let \(f\geq 0\) be a measurable function. For \(k\in\mathbb{Z}^{+}\), \(k\neq 0\) divides the interval \([0,k]\) into subintervals

\[\left[\frac{j-1}{2^{k}},\frac{j}{2^{k}}\right]\quad\text{where}\quad j=1,2, \ldots,k2^{k}.\]