Math Problems Vault

Problem 4.37

Prove the theorem regarding the continuity of integration.

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Problem 4.38

Let $f:[0, \infty ] \rightarrow \mathbb {R}$ be integrable. Show that

\lim _{n \rightarrow \infty } \int _{1 / n}^{n} f d m = \int _{0}^{\infty } f d m

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Problem 4.39

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be integrable. Show that the function $F: \mathbb {R} \rightarrow \mathbb {R}$ defined by

F(x) = \int _{-\infty }^{x} f d m \text{ for all } x \in \mathbb {R}

is properly defined and continuous. Is it Lipschitz?

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Problem 4.40

(i)(ii)

For each of the two functions $f$ on $[1, \infty )$ defined below, show that $\lim_{n \rightarrow \infty } \int_{[1, n]} f d m$ exists while $f$ is not integrable over $[1, \infty )$ . Does this contradict the continuity of integration?

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(i)

Define $f(x) = (-1)^{k} / k$ , for $k \leq x < k+1$ .

(ii)

Define $f(x) = (\sin x) / x$ for $1 \leq x < \infty$ .

Problem 4.41

(i)(ii)(iii)(iv)

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be integrable. Show that the following four assertions are equivalent:

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(i)

$f = 0$ almost everywhere on $\mathbb {R}$ .

(ii)

$\int_{\mathbb {R}} f g d m = 0$ for every bounded, finitely supported, measurable function $g$ .

(iii)

$\int_{A} f d m = 0$ for every measurable set $A$ .

(iv)

$\int_{\mathcal{O}} f d m = 0$ for every open set $\mathcal{O}$ .

Problem 4.42

(i)(ii)(iii)

Let $f:[a, b] \rightarrow \mathbb {R}$ be integrable and $\epsilon > 0$ . Establish the following approximations.

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(i)

There is a simple function $\eta :[a, b] \rightarrow \mathbb {R}$ for which $\int_{[a, b]}\left|f-\eta \right| d m < \epsilon$ .

(ii)

There is a step-function $s:[a, b] \rightarrow \mathbb {R}$ for which $\int_{[a, b]}\left|f-s\right| d m < \epsilon$ .

(iii)

There is a continuous function $g:[a, b] \rightarrow \mathbb {R}$ for which $\int_{[a, b]}\left|f-g\right| d m < \epsilon$ .