Measure Theory/Approximations of Sequences of Functions

Approximations of Sequences of Functions

The content of this lesson is the final of Littlewood's three principles, which are also known as Egoroff's Theorem and Lusin's Theorem.

Pointwise Convergence to Nearly Uniform

Here we will show that pointwise convergence can, through an appropriate domain restriction, entail something like uniform convergence.

There are a few theorems that show something like the rough expression above. The first interpretation that we will prove, in a slightly more formal expression, is:

For $\delta$ as small as you like, you never need to remove more than $100\delta$ percent of the domain, for the convergence to be uniform.

The formal expression of this is: Let

• $E\in {\mathcal {M}}$ has finite measure,
• $\langle f_{n}\rangle$ a sequence of measurable functions defined on E,
• $f_{n}\to f$ pointwise on E,

Then $\forall \delta ,\varepsilon \in {\mathbb {R}}^{+},\quad \exists F\in {\mathcal {M}},\quad \exists N\in {\mathbb {Z}}^{+}$ such that

• $F\subseteq E$
• on $E\smallsetminus F,\quad |f_{n}(x)-f(x)|<\varepsilon \quad \forall n\geq N$
• $\lambda (F)<\delta$

To begin the proof, let $\varepsilon ,\delta \in {\mathbb {R}}^{+}$.

With the assumptions as above, define for each $1\leq n$ the set of points where the sequence is "far apart":

$G_{n}=\{x\in E:|f_{n}(x)-f(x)|\geq \varepsilon \}$

Exercise 1. Where the Sequence Is Far

Definition: sup and limsup

Let $\langle A_{n}\rangle$ be any sequence of sets, each a subset of some set X. Define the supremum of $\langle A_{n}\rangle$ to be their union. Also define the limit superior of $\langle A_{n}\rangle$ to be the intersection over all suprema of tails:

${\overline {\lim }}_{k}A_{k}=\bigcap _{N=1}^{\infty }\bigcup _{N\leq k}A_{k}$.

Of course every definition above for suprema has a correlate for infima.

Part A. Prove that the subsets of any set X are well-ordered by the subset relation.  

Part B. Prove that the union of $\langle A_{n}\rangle$ is an upper bound on this sequence, for the subset relation.  Also, and in the same sense, prove that it is the least of the upper bounds.

Part C. Explain why the limit supremum for real numbers is analogous to the limit supremum for sets.  There are several equivalent definitions of the limit supremum for a sequence of real numbers, $\langle a_{k}\rangle$.  So take the definition to be 

${\overline {\lim }}a_{k}=\lim _{N\to \infty }\sup\{a_{k}|N\leq k\}$

Part D. State the definitions that make the most sense for infima and limit inferiors, for sets.  

Part E. Now prove that the limit superior of $G_{n}$ is the empty set.

Part F. Define the union of the Nth tail, $E_{N}=\bigcup _{N\leq n}G_{n}$ and apply the continuity of measure, to infer that $\exists N\in {\mathbb {Z}}^{+}$ such that $\lambda (E_{N})<\delta$

Part G. Conclude the rest of the proof of the theorem.

Part H.

Definition: converge a.e.

Let $E\in {\mathcal {M}}$ and $\langle f_{n}\rangle$ be a sequence of measurable functions defined on E. Let $F\subseteq E$ be a measurable subset, such that $\lambda (F)=\lambda (E)$ and $f_{n}(x)\to f(x)$ for all $x\in F$. Then we say that $f_{n}\to f$ pointwise a.e.

Show that the result we finished proving in Part G. also holds if the condition of pointwise convergence is replaced by pointwise a.e. convergence.

Egoroff's Theorem

Egoroff's theorem states the following.

Let $\langle f_{n}\to \rangle$ pointwise a.e. on a measurable set E of finite measure.

Then $\forall \eta \in {\mathbb {R}}^{+}\quad \exists (F\subseteq E)$ such that

• $\lambda (F)<\eta$
• $f_{n}{\overset {u}{\to }}f$ on $E\smallsetminus F$.

Exercise 2. Prove Egoroff's

Prove Egoroff's theorem.

Hint: Use the Part H. of Exercise 1. Where the Sequence Is Far together with the $\eta /2^{n}$ trick.

Lusin's Theorem

Lusin's theorem states the following.

Let

• $f:[a,b]\to {\mathbb {R}}$ be a measurable real-valued function.
• $\varepsilon \in {\mathbb {R}}^{+}$

Then there exists a continuous $g:[a,b]\to {\mathbb {R}}$ such that $\lambda (\{x\in [a,b]:f(x)\neq g(x)\})<\delta$.

Exercise 3. Prove Lusin's

Prove Lusin's theorem.

Hint: Use Egoroff's and the approximation of measurable functions by continuous functions.