Sublevel Set Estimates

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Problem 2E.14 from Axler S. "Measure, Integration & Real Analysis" [Axl19]More problems in Section 2E "Convergence of Measurable Functions"

Suppose b1,b2,…b_{1}, b_{2}, \ldots is a sequence of real numbers. Define f:R→[0,∞]f: \mathbb {R} \rightarrow [0, \infty ] by

f(x)={∑k=1∞14k∣x−bk∣ if x∉{b1,b2,…}∞ if x∈{b1,b2,…} f(x) = \begin{cases} \sum _{k = 1}^{\infty } \frac{1}{4^{k}\left|x-b_{k}\right|} & \text{ if } x \notin \left\{ b_{1}, b_{2}, \ldots \right\} \\ \infty & \text{ if } x \in \left\{ b_{1}, b_{2}, \ldots \right\} \end{cases}

Prove that ∣{x∈R:f(x)<1}∣=∞\left|\left\{ x \in \mathbb {R}: f(x) < 1\right\} \right| = \infty.

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Remark.
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This exercise is a variation of a problem originally considered by Borel. If b1,b2,…b_{1}, b_{2}, \ldots contains all the rational numbers, then it is not even obvious that {x∈R:f(x)<∞}≠∅\left\{ x \in \mathbb {R}: f(x) < \infty \right\} \neq \varnothing.

Problem VIII.2.18 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section VIII.2 "Measurable Functions"

Let the function ff be nn times continuously differentiable on an interval II and let ∣f(n)(x)∣⩾Δ>0\left|f^{(n)}(x)\right| \geqslant \Delta >0 for all x∈Ix \in I. Prove that

λ{x∈I∣∣f(x)∣⩽t}⩽2n2(tΔ)1/n. \lambda \left\{ x \in I \mid \left|f(x)\right| \leqslant t\right\} \leqslant 2 n^{2}\left(\frac{t}{\Delta }\right)^{1 / n} .
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References

[Axl19] Axler S. Measure, Integration & Real Analysis. 3rd ed., Springer, 2019.
[МГЛ+04] Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. Избранные задачи по вещественному анализу. 2-е изд., Невский Диалект; БХВ-Петербург, 2004.