Graph of measurable function is measurable (and has measure zero)

Let f be a finite real valued measurable function on a measurable set E\subseteq\mathbb{R}. Show that the set \{(x,f(x)):x\in E\} is measurable.

We define \Gamma(f,E):=\{(x,f(x)):x\in E\}. This is popularly known as the graph of a function. Without loss of generality, we may assume that f is nonnegative. This is because we can write f=f^+ - f^-, where we split the function into two nonnegative parts.

The proof here can also be found in Wheedon’s Analysis book, Chapter 5.

The strategy for proving this question is to approximate the graph of the function with arbitrarily thin rectangular strips. Let \epsilon>0. Define E_k=\{x\in E\mid \epsilon k\leq f(x)<\epsilon (k+1)\}, k=0,1,2,\dots.

We have |\Gamma (f,E_k)|_e\leq\epsilon |E_k|, where |\cdot|_e indicates outer measure.

Also, \Gamma(f,E)=\cup\Gamma(f,E_k), where \Gamma(f,E_k) are disjoint.

\begin{aligned}|\Gamma(f,E)|_e&\leq\sum_{k=1}^\infty|\Gamma(f,E_k)|_e\\    &\leq\epsilon(\sum_{k=1}^\infty|E_k|)\\    &=\epsilon|E|    \end{aligned}

If |E|<\infty, we can conclude |\Gamma(f,E)|_e=0 and thus \Gamma(f,E) is measurable (and has measure zero).

If |E|=\infty, we partition E into countable union of sets F_k each with finite measure. By the same analysis, each \Gamma(f,F_k) is measurable (and has measure zero). Thus \Gamma(f,E)=\bigcup_{k=1}^\infty\Gamma(f,F_k) is a countable union of measurable sets and thus is measurable (has measure zero).

Measure Theory: What does a.e. (almost everywhere) mean

Source: Elements of Integration by Professor Bartle

Students studying Mathematical Analysis, Advanced Calculus, or probability would sooner or later come across the term a.e. or “almost everywhere”.

In layman’s terms, it means that the proposition (in the given context) holds for all cases except for a certain subset which is very small. For instance, if f(x)=0 for all x, and g(x)=0 for all nonzero x, but g(0)=1, the function f and g would be equal almost everywhere.

For formally, a certain proposition holds \mu-almost everywhere if there exists a subset N\in \mathbf{X} with \mu (N)=0 such that the proposition holds on the complement of N. \mu is a measure defined on the measure space \mathbf{X}, which is discussed in a previous blog post: What is a Measure.

Two functions f, g are said to be equal \mu-almost everywhere when f(x)=g(x) when x\notin N, for some N\in X with \mu (N)=0. In this case we would often write f=g, \mu-a.e.

Similarly, this notation can be used in the case of convergence, for example f=\lim f_n, \mu-a.e.

The idea of “almost everywhere” is useful in the theory of integration, as there is a famous Theorem called “Lebesgue criterion for Riemann integrability”.

(From Wikipedia)

A function on a compact interval [ab] is Riemann integrable if and only if it is bounded and continuous almost everywhere (the set of its points of discontinuity has measure zero, in the sense of Lebesgue measure). This is known as the Lebesgue’s integrability condition or Lebesgue’s criterion for Riemann integrability or the Riemann—Lebesgue theorem.[4] The criterion has nothing to do with the Lebesgue integral. It is due to Lebesgue and uses his measure zero, but makes use of neither Lebesgue’s general measure or integral.

Reference book:

Functions between Measurable Spaces

Sometimes, it is  desirable to define measurability for a function f from one measurable space (X,X) into another measurable space (Y,Y). In this case one can define f to be measurable if and only if the set f^{-1} (E)=\{x\in X: f(x) \in E\} belongs to X for every set E belonging to Y.

This definition of measurability appears to differ from Definition 2.3 (earlier in the book), but Definition 2.3 is in fact equivalent to this definition in the case that Y=R and Y=B.

First, lets recap what is Definition 2.3:

A function f on X to R is said to be X-measurable (or simply measurable) if for every real number \alpha the set \{x\in X:f(x)>\alpha\} belongs to X.

Let (X,X) be a measurable space and f be a real-valued function defined on X. Then f is X-measurable if and only if f^{-1 }(E)\in X for every Borel set E.

Thus there is a close analogy between the measurable functions on a measurable space and continuous functions on a topological space.

Source:

The Elements of Integration and Lebesgue Measure

Measurability of product fg

In the previous chapters, Bartle showed that that if f is in M(X,X), then the functions cf, f^2, |f|, f^+, f^- are also in M(X,X).

The case of the measurability of the product fg when f, g belong to M(X,X) is a little bit more tricky. If n\in\mathbb{N}, let f_n be the “truncation of f” defined by f_n (x)=\begin{cases}f(x), &\text{if }|f(x)|\leq n, \\ n, &\text{if } f(x)>n,\\ -n, &\text{if }f(x)<-n\end{cases}

Let g_m be defined similarly. We will work out the proof that f_n and g_m are measurable (Bartle left it as Exercise 2.K).

Proof:

Each f_n is a function on X to \mathbb{R}.

\{x\in X:f_n (x) >\alpha\}=\begin{cases}\{x \in X: f(x)>\alpha\}, &\text{if }-n<\alpha <n,\\ \emptyset, &\text{if }\alpha\geq n,\\X, &\text{if }\alpha\leq -n \end{cases}

All of the above sets are in X.

Thus, we may use an earlier Lemma 2.6 to show that the product f_n g_m is measurable.

We also have f(x)g_m (x)=\lim_n f_n (x)g_m (x), and using an earlier corollary that says that if a sequence (f_n) is in M(X,X) converges to f on X, then f is also in M(X,X), we have that f(x)g_m (x) belongs to M(X,X).

Finally, (fg)(x)=f(x)g(x)=\lim_m f(x)g_m (x), and hence fg also belongs to M(X,X).

This is a very powerful result of Lebesgue integration, since we can see that the theory includes extended real-valued functions, and prepares us to integrate functions that can reach infinite values!

Source: The Elements of Integration and Lebesgue Measure

Measure and Integration Recommended Book

I have added a new addition to the Recommended Books for Undergraduate Math, which is one of my most popular posts!

The new book is The Elements of Integration and Lebesgue Measure, an advanced text on the theory of integration. At the high school level, students are exposed to integration, but merely the rules of integration. At university, students learn the Riemann theory of integration (Riemann sums), which is a good theory, but not the best. There are some functions which we would like to integrate, but do not fit nicely into the theory of Riemann Integration.

I am personally reading this book as well, as I didn’t manage to study it in university, but it is a key component for graduate level analysis. Students interested in advanced Probability (see this post on Coursera Probability course) would be needing Lebesgue theory too!