# Understanding Lusin's Theorem for $\mathbb{R}$ and Its Proof

• MHB
• joypav
In summary, the problem involves showing the existence of a sequence of continuous functions that converge point wise to a given measurable function. This is similar to Lusin's Theorem for the real numbers and the proof involves applying the Approximation Theorem and using Egorov's Theorem to find a closed set where the function is continuous. By considering a set of open, disjoint intervals, the continuous function can be extended to the entire real line.
joypav
Problem:
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be measurable. Then there exists a sequence of continuous functions $(g_n)$ such that $limg_n(x)$ exists for all $x \in \mathbb{R}$ and $limg_n(x) = f(x)$ a.e. x.

Is this like Lusin's Theorem? Lusin's theorem for the real numbers? If so, how does this change the proof?

Here is my proof (posting for completeness but if you have corrections feel free to comment!).

Proof:
Apply the Approximation Theorem to $f$.
Let $(f_n)$ be a sequence of simple functions such that $f_n \rightarrow f$ point wise.
For every $n \in \Bbb{N}$, let
$F_n \subset E$ be closed such that $m(R - F_n) < \frac{\epsilon}{2^k}$ such that $f_n |_{F_n}$ is continuous.
(By Egorov's) $\implies \exists B \subset R$ such that $m(B) < \epsilon$ and $f_n \rightarrow f$ uniformly on $R - B$.
WLOG, assume $B$ is open.
Then $R - B$ is closed.
Let $F_0 = R - B$.

Claim: $f |_{F_\infty}$ is continuous.
A uniformly convergent sequence of continuous functions is continuous
$\implies f |_{F_\infty} = F_0 \cap \cap_{k}F_k$ is continuous.
Then,
$m(E - F_\infty) \leq \sum_{n=0}^{\infty}m(E - F_n) < 2\epsilon$
$\implies$ We've found $F \subset R$ closed such that $m(R - F) < \epsilon$ and $f |_F$ is continuous.

Now, consider our set F.
$\exists G_1, G_2,...$ open and pairwise disjoint such that $F^c = \cup_{i \in \Bbb{N}}G_i$.

Let $G_i = (a_i, b_i)$. Let,
$g(x) =$
$f(x), x \in F$
$(\frac{b_i - x}{b_i - a_i})f(a_i) + (\frac{x - a_i}{b_i - a_i})f(b_i), x \in (a_i, b_i)$

Then,
$f |_F$ continuous $\implies g |_F$ is continuous.
We extend $g$ so that it is also continuous on $F^c$ by the definition for $g$ given.
We conclude that,
$g : R \rightarrow R$ is continuous and $g |_F = f$.

Yes, this problem is similar to Lusin's Theorem for the real numbers. In fact, Lusin's Theorem states that for any measurable function $f: \mathbb{R} \rightarrow \mathbb{R}$, there exists a sequence of continuous functions $(g_n)$ such that $g_n \rightarrow f$ pointwise on $\mathbb{R}$.

The main difference between this problem and Lusin's Theorem is the use of the phrase "a.e. x" (almost everywhere). This means that the sequence of continuous functions $(g_n)$ only needs to converge to $f$ for almost every point $x$ in $\mathbb{R}$, rather than for every point as in Lusin's Theorem.

To incorporate this into the proof, we can use the fact that a measurable function can be approximated by a continuous function on a set of arbitrarily small measure. This allows us to construct a sequence of continuous functions that converges to $f$ on a set of measure 0, which satisfies the condition of convergence almost everywhere.

## 1. What is Lusin's Theorem for $\mathbb{R}$?

Lusin's Theorem for $\mathbb{R}$ is a mathematical theorem that states that every measurable function on the real line can be approximated by a continuous function on a subset of the real line with arbitrarily small error. In other words, it shows that measurable functions can be "smoothed out" on a subset of the real line.

## 2. What is the significance of Lusin's Theorem for $\mathbb{R}$?

Lusin's Theorem for $\mathbb{R}$ has many applications in mathematics, particularly in integration theory and functional analysis. It allows for a better understanding of the behavior of measurable functions and their relationship to continuous functions. It also plays a crucial role in the proof of other important theorems, such as the Lebesgue Differentiation Theorem.

## 3. What is the proof of Lusin's Theorem for $\mathbb{R}$?

The proof of Lusin's Theorem for $\mathbb{R}$ involves constructing a sequence of continuous functions that converge to the original measurable function. This is done by using the properties of the Lebesgue measure and approximating the function on smaller and smaller subsets of the real line. The details of the proof can be quite technical and involve measure theory and topological concepts.

## 4. Are there any variations of Lusin's Theorem for $\mathbb{R}$?

Yes, there are several variations of Lusin's Theorem for $\mathbb{R}$ that involve different types of functions and domains. For example, there are versions for functions defined on higher-dimensional spaces or on more general measure spaces. There are also variations that involve different types of approximations, such as polynomial or trigonometric approximations.

## 5. Are there any real-world applications of Lusin's Theorem for $\mathbb{R}$?

While Lusin's Theorem for $\mathbb{R}$ may seem abstract and theoretical, it has many practical applications in fields such as physics, engineering, and economics. For example, it can be used to model and analyze physical processes that involve continuous functions, such as heat flow or fluid dynamics. It can also be applied in signal processing to reconstruct noisy signals or in data analysis to smooth out noisy data.

• Topology and Analysis
Replies
2
Views
4K
• Topology and Analysis
Replies
4
Views
245
• Topology and Analysis
Replies
3
Views
485
• Topology and Analysis
Replies
2
Views
1K
• Topology and Analysis
Replies
4
Views
698
• Topology and Analysis
Replies
7
Views
2K
• Topology and Analysis
Replies
1
Views
361
• Topology and Analysis
Replies
2
Views
2K
• Topology and Analysis
Replies
8
Views
379
• Topology and Analysis
Replies
2
Views
1K