Measure Zero Sets, Oscillation, and Lebesgue's Criterion Review

We will now review some of the recent content posted regarding subsets of $\mathbb{R}$ with measure $0$, the oscillation of a bounded function on a subset of a closed interval (and at a point in that closed interval), and the extremely important Lebesgue's criterion.

• On the Subsets of Real Numbers with Measure Zero page we said that a subset $S \subset \mathbb{R}$ is said to have Measure Zero denoted $m(S) = 0$ if for all $\epsilon > 0$ there exists a countable open interval covering $\{ I_k = (a_k, b_k) : k \in K \}$ (where $K$ is some countable indexing set), i.e., $\displaystyle{S \subseteq \bigcup_{K} I_k}$ and such that if $l(I_k) = b_k - a_k$ (the length of each open interval in the countable covering) then:

(1)

\begin{align} \quad \sum_{K} l(Ik) = \sum_{K} (b_k - a_k) < \epsilon \end{align}

• We noted that any finite subset $S = \{ x_1, x_2, ..., x_n \} \subset \mathbb{R}$ has measure $0$.

• On The Measure of Countable Subsets of Real Numbers page we went further and showed that any countable subset of the real numbers also has measure $0$.

• On The Measure of a Countable Collection of Measure Zero Subsets of Real Numbers page we saw that if $\{ F_1, F_2, ..., F_n, ... \}$ is an infinitely countable collection of subsets of $\mathbb{R}$ with measure $0$ then the union $F = \bigcup_{j=1}^{\infty} F_j$ is subset of $\mathbb{R}$ with measure $0$.

• Later on, we saw from the Oscillation of a Bounded Function on a Set page that if $f$ is a bounded function on $[a, b]$ and $T \subseteq [a, b]$ then the Oscillation of $f$ on $T$ is denoted $\Omega_f (T)$ and is defined as:

(2)

\begin{align} \quad \Omega_f (T) = \sup \{ f(x) - f(y) : x, y \in T \} \end{align}

• We saw that $\Omega_f (T)$ is always nonnegative and well defined and always exists for a bounded function $f$ any any subset $T \subseteq [a, b]$ since if $f$ is bounded on $[a, b]$ we have that $\mid f(x) \mid \leq M$ for some $M \in \mathbb{R}$, $M > 0$ and so $f$ is also bounded on any $T \subseteq [a, b]$ and $f(x) - f(y) \leq \mid f(x) - f(y) \mid \leq \mid f(x) \mid + \mid f(y) \mid \leq 2M$.

• We noted that the oscillation of a singleton set $\{ c \} \subseteq [a, b]$ is equal to $0$.

• On the Oscillation of a Bounded Function at a Point page we looked at a similar concept. For a bounded function $f$ on $[a, b]$ and for $x \in [a, b]$ we defined the Oscillation of $f$ at $x$ denoted $\omega_f (x)$ to be:

(3)

\begin{align} \quad \omega_f (x) = \lim_{h \to 0} \Omega_f ((x - h, x + h) \cap [a, b]) \end{align}

• We then proved an extremely important theorem on the Oscillation and Continuity of a Bounded Function at a Point page. We saw that if $f$ is a bounded function on $[a, b]$ and $c \in [a, b]$ then $f$ is continuous at $c$ if and only if $\omega_f (c) = 0$.

• We then got into the main theorem of this section. On the Lebesgue's Criterion Part 1 - Riemann Integrability of a Bounded Function and Lebesgue's Criterion Part 2 - Riemann Integrability of a Bounded Function pages we looked at Lebesgue's Criterion for the Riemann integrability of a bounded function. We saw that if $f$ is a bounded function on $[a, b]$ and $D$ is the set of discontinuities of $f$ on $[a, b]$ then $f$ is Riemann integrable if and only the measure of the set of discontinuities of $f$ on $[a, b]$ is equal to $0$, that is, $m(D) = 0$.

• Many nice results, some of which have already been proven, were then reproved on the Corollaries to Lebesgue's Criterion for the Riemann Integrability of a Bounded Function using Lebesgue's criterion. For example, we reproved that every function of bounded variation on $[a, b]$ is Riemann integrable on $[a, b]$.

• We also saw that if $f$ and $g$ were bounded functions on $[a, b]$ whose sets of discontinuities on $[a, b]$, $D_f$ and $D_g$ respectively are equal $(D_f = D_g)$ then $f$ is Riemann integrable on $[a, b]$ if and only if $g$ is Riemann integrable on $[a, b]$.