Math Problems Vault

Problem 4B.1

For $f \in \mathcal{L}^{1}(\mathbb {R})$ and $I$ an interval of $\mathbb {R}$ with $0 < \left|I\right| < \infty$ , let $f_{I}$ denote the average of $f$ on $I$ . In other words, $f_{I} = \frac{1}{\left|I\right|} \int_{I} f$ .

Suppose $f \in \mathcal{L}^{1}(\mathbb {R})$ . Prove that

\lim _{t \downarrow 0} \frac{1}{2 t} \int _{b-t}^{b+t}\left|f-f_{[b-t, b+t]}\right| = 0

for almost every $b \in \mathbb {R}$ .

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Problem 4B.2

For $f \in \mathcal{L}^{1}(\mathbb {R})$ and $I$ an interval of $\mathbb {R}$ with $0 < \left|I\right| < \infty$ , let $f_{I}$ denote the average of $f$ on $I$ . In other words, $f_{I} = \frac{1}{\left|I\right|} \int_{I} f$ .

Suppose $f \in \mathcal{L}^{1}(\mathbb {R})$ . Prove that

\lim _{t \downarrow 0} \sup \left\{ \frac{1}{\left|I\right|} \int _{I}\left|f-f_{I}\right|: I \text{ is an interval of length } t \text{ containing } b\right\} = 0

for almost every $b \in \mathbb {R}$ .

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Problem 4B.3

Suppose $f: \mathbb {R} \rightarrow \mathbb {R}$ is a Lebesgue measurable function such that $f^{2} \in \mathcal{L}^{1}(\mathbb {R})$ . Prove that

\lim _{t \downarrow 0} \frac{1}{2 t} \int _{b-t}^{b+t}\left|f-f(b)\right|^{2} = 0

for almost every $b \in \mathbb {R}$ .

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Problem 4B.4

Prove that the Lebesgue Differentiation Theorem (4.19) still holds if the hypothesis that $\int_{-\infty }^{\infty }\left|f\right| < \infty$ is weakened to the requirement that $\int_{-\infty }^{x}\left|f\right| < \infty$ for all $x \in \mathbb {R}$ .

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Remark.

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4.19 (Lebesgue Differentiation Theorem, second version): Suppose $f \in \mathcal{L}^{1}(\mathbb {R})$ . Define $g: \mathbb {R} \rightarrow \mathbb {R}$ by $g(x) = \int_{-\infty }^{x} f$ . Then $g^{\prime }(b) = f(b)$ for almost every $b \in \mathbb {R}$ .

Problem 4B.5

Suppose $f: \mathbb {R} \rightarrow \mathbb {R}$ is a Lebesgue measurable function. Prove that

\left|f(b)\right| \leq f^{*}(b)

for almost every $b \in \mathbb {R}$ .

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Problem 4B.6

Prove that if $h \in \mathcal{L}^{1}(\mathbb {R})$ and $\int_{-\infty }^{s} h = 0$ for all $s \in \mathbb {R}$ , then $h(s) = 0$ for almost every $s \in \mathbb {R}$ .

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Problem 4B.7

Give an example of a Borel subset of $\mathbb {R}$ whose density at 0 is not defined.

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Problem 4B.8

Give an example of a Borel subset of $\mathbb {R}$ whose density at 0 is $\frac{1}{3}$ .

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Problem 4B.9

Prove that if $t \in [0,1]$ , then there exists a Borel set $E \subseteq \mathbb {R}$ such that the density of $E$ at 0 is $t$ .

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Problem 4B.10

Suppose $E$ is a Lebesgue measurable subset of $\mathbb {R}$ such that the density of $E$ equals 1 at every element of $E$ and equals 0 at every element of $\mathbb {R} \backslash E$ . Prove that $E = \varnothing$ or $E = \mathbb {R}$ .

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