The Integral

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space and $s$ a simple function on $X$. If $E$ is a measurable set of measure 0 , prove that $\int_{E} s d \mu =0$.

Prove that if $E \subset X$ is measurable then $\int_{E} f d \mu =\int_{X} f \chi_{E} d \mu$.

Prove Proposition 7.2.11.

Proposition 7.2.11: Let $(X, \mathfrak {M}, \mu )$ be a measure space and $f, g$ nonnegative measurable functions on $X$. (i) If $\alpha \geq 0$, then $\int_{X} \alpha f \, d\mu = \alpha \int_{X} f \, d\mu$. (ii) If $f \leq g$ a.e. on $X$, then $\int_{X} f \, d\mu \leq \int_{X} g \, d\mu$. (iii) If $X_{0}$ is a measurable subset of $X$ and $\mu \left(X \setminus X_{0}\right) = 0$, then $\int_{X} f \, d\mu = \int_{X_{0}} f \, d\mu$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space and $f$ a nonnegative measurable function on $X$. If $A, B$ are disjoint measurable subsets of $X$, prove that $\int_{A \cup B} f d \mu =\int_{A} f d \mu +\int_{B} f d \mu$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space and $f$ a nonnegative, bounded, measurable function on $X$. Prove that $f \in L(X, \mu )$ if and only if the series

converges.

Let ( $X, \mathfrak {M}, \mu$ ) be a finite measure space and $f$ a nonnegative, bounded and measurable function on $X$. Prove that

Does the assertion remain true if $\mu (X)=\infty$ ?

Let $\lambda$ be a set function on all subintervals $[a, b] \subset [0,1]$ defined by $\lambda ([a, b])=\varphi (b)-\varphi (a)$, where $\varphi$ is the Cantor-Lebesgue function on $[0,1]$. Let $\mu$ be the Caratheodory extension of $\lambda$ to $\mathcal{M} \mid [0,1]$. Calculate $\int_{0}^{1} x d \mu$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space and $\left\{ f_{n}\right\}$ a sequence of nonnegative measurable functions on $X$ that converges a.e. to a function $f$. Prove or disprove: The limit

exists (finite or infinite).

Find a sequence of real valued nonnegative function $f_{n}$ on $[0,1]$ so that

and

Let $E$ be a subset of $\mathbb {R}$ that is of finite Lebesgue measure and let $f$ be a measurable function on $E$ that is a.e. strictly positive. Prove that for every $0<\alpha <m(E)$

Prove Corollary 7.2.15.

Corollary 7.2.15: Let $(X, \mathfrak {M}, \mu )$ be a measure space and $\left\{ f_{n}\right\}$ a sequence of nonnegative measurable functions on $X$. Then $\int_{X} \sum_{n=1}^{\infty } f_{n} \, d\mu = \sum_{n=1}^{\infty } \int_{X} f_{n} \, d\mu$.

Let $(X, \mathfrak {M}, \mu )$ be a measure space and $f: X \rightarrow [0,+\infty )$ a measurable function. Prove that $f \in L(X, \mu )$ if and only if $\sup \int_{A} f d \mu <\infty$, where the supremum is taken over all sets $A$ of finite measure on which $f$ is bounded above.

Let $(X, \mathfrak {M}, \mu )$ be a measure space, let $\left\{ A_{n}\right\} \subset \mathfrak {M}$ such that $\mu \left(\cup_{n=1}^{\infty } A_{n}\right)$ is finite, and let $f$ be a measurable function that is strictly positive a.e. and such that $\lim_{n} \int_{A_{n}} f d \mu =0$. Prove that $\lim_{n} \mu \left(A_{n}\right)=0$.

Let ( $X, \mathfrak {M}, \mu$ ) be a finite measure space. Prove that a nonnegative measurable function $f$ is integrable over $X$ if and only if

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space, and let $f$ be a nonnegative measurable function that is finite a.e., and let $\left\{ f_{n}\right\}$ be defined by

Prove that the sequence $\left\{ \int_{X} f_{n} d \mu \right\}$ is monotone increasing and that it converges to $\int_{X} f d \mu$. Will the assertion still hold if $f$ is not assumed to be finite a.e.?

Let $\left\{ f_{n}\right\}$ be a sequence of nonnegative Lebesgue measurable functions on $(0,10)$ that converges to a function $f$ for almost all $x \in (0,10)$. Let $F(x)= \int_{0}^{x} f d m$ and $F_{n}(x)=\int_{0}^{x} f_{n} d m$. Prove that

Show that the Monotone Convergence Theorem may not hold for decreasing sequences of functions.

Let $\left\{ f_{n}\right\}$ be a decreasing sequence of nonnegative integrable functions on $(X, \mathfrak {M}, \mu )$ that converges to $f$ a.e. Prove that $\lim f_{n}=0$ a.e. if and only if $\lim \int f_{n} d \mu =0$.

Show that the Monotone Convergence Theorem holds for a decreasing sequence of functions $\left\{ f_{n}\right\}$ if $f_{1} \in L(X, \mu )$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space and $f, g$ be nonnegative functions in $L(X, \mu )$ and define

Prove that

Let $\left\{ f_{n}\right\}$ be a sequence of nonnegative measurable functions that converge pointwise to a function $f$ on $\mathbb {R}$. Suppose that

Prove that

for every measurable set $A$.

Let $f$ be a function defined on $[0,1]$ in the following way. If $x$ belongs to the Cantor set, then $f(x)=0$. If $x$ belongs to a complementary interval of length $3^{-k}$, then $f(x)=k$. Find $\int_{[0,1]} f$.

Let $\mu$ and $\nu$ be measures on the measurable space ( $X, \mathfrak {M}$ ), let $\lambda =\mu +\nu$, and let $f$ be a nonnegative measurable function on $X$. Prove that, for any $E \in \mathfrak {M}$,

Let $\left\{ r_{n}\right\}$ be an enumeration of rational numbers in $(0,1)$, let $Q_{n}= \left\{ r_{1}, r_{2}, \ldots , r_{n}\right\}$, and let

Prove that each $f_{n}$ is measurable and find

Let ( $X, \mathfrak {M}, \mu$ ) be a finite measure space and let $f$ be a measurable, nonnegative function on $X$ such that $f^{n} \in L(X, \mu )$, for all $n \in \mathbb {N}$.

(Beppo Levi's Theorem.) Let ( $X, \mathfrak {M}, \mu$ ) be a measure space and let $\left\{ f_{n}\right\}$ be an increasing sequence of nonnegative measurable functions such that $\sup_{n} \int_{X} f_{n} d \mu <\infty$. Prove that

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space and let $E$ be a set of measure 0 . Suppose that $f=\infty$ on $E$. Prove that $\int_{E} f=0$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space, let $\mathfrak {M}_{0}$ be a $\sigma$-algebra contained in $\mathfrak {M}$, let $\mu_{0}$ be the restriction of $\mu$ to $\mathfrak {M}_{0}$, and let $f$ be a nonnegative function that is measurable with respect to $\mathfrak {M}_{0}$. Prove that $f$ is measurable with respect to $\mathfrak {M}$ and

Can the inequality be strict?

Prove or disprove: there exists a nonnegative $f \in L(\mathbb {R})$ such that for any $a<b$ and any $M>0$ the set $(a, b) \cap \left\{ x \in \mathbb {R}: f(x) \geq M\right\}$ has positive measure.

Let $(X, \mathfrak {M}, \mu )$ be a measure space and let $f$ be a nonnegative function that is integrable over $X$. Prove that for every $\varepsilon >0$ there exists a set $A \in \mathfrak {M}$ such that $\mu (A)<\infty$ and $\int_{A} f>\int_{X} f-\varepsilon$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space, let $f$ be a nonnegative function that is integrable over $X$, and let $\varepsilon >0$. Prove that there exists a set $A \in \mathfrak {M}$ such that $\mu (A)<\infty$ and a simple function $s$ that vanishes outside of $A$ and satisfies $0 \leq s \leq f$ and $\int_{X}|f-s| d \mu <\varepsilon$.

Prove the Monotone Convergence Theorem under a weaker assumption that $\lim f_{n}=f$ in measure (instead of a.e.).

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space, let $f$ be a nonnegative function on $X$, and let $\left\{ X_{n}\right\}$ be an increasing sequence of measurable sets such that $X=\cup_{n=1}^{\infty } X_{n}$. Prove that $f$ is integrable over $X$ if and only if the sequence $\left\{ \int_{X_{n}} f d \mu \right\}$ is bounded.

Let $(X, \mathfrak {M}, \mu )$ be a finite measure space. For any measurable functions $f, g$ on $X$ we define

Prove that $d$ has all properties of distance except that $d(f, g)=0$ does not imply that $f=g$.

Let $(X, \mathfrak {M}, \mu )$ be a finite measure space and, as in Exercise 7.2.34, for measurable functions $f, g$ on $X$ let $d(f, g)=\int_{X} \frac{|f-g|}{1+|f-g|} d \mu$. Exercise 7.2.34 motivates the construction of a metric space $M(X, \mu )$ whose elements are equivalence classes of measurable functions. ( $f \sim g$ if and only if $d(f, g)=0$.) Assuming that ( $X, \mathfrak {M}, \mu$ ) is a finite measure space, this is indeed a metric space. In Exercises 7.2.35-7.2.38, we will explore some of its properties.

Prove that convergence in the space $M(X, \mu )$ is equivalent to the convergence in measure.

Let $(X, \mathfrak {M}, \mu )$ be a finite measure space and, as in Exercise 7.2.34, for measurable functions $f, g$ on $X$ let $d(f, g)=\int_{X} \frac{|f-g|}{1+|f-g|} d \mu$. Exercise 7.2.34 motivates the construction of a metric space $M(X, \mu )$ whose elements are equivalence classes of measurable functions. ( $f \sim g$ if and only if $d(f, g)=0$.) Assuming that ( $X, \mathfrak {M}, \mu$ ) is a finite measure space, this is indeed a metric space. In Exercises 7.2.35-7.2.38, we will explore some of its properties.

Prove that the space $M(X, \mu )$ is complete.

Let $(X, \mathfrak {M}, \mu )$ be a finite measure space and, as in Exercise 7.2.34, for measurable functions $f, g$ on $X$ let $d(f, g)=\int_{X} \frac{|f-g|}{1+|f-g|} d \mu$. Exercise 7.2.34 motivates the construction of a metric space $M(X, \mu )$ whose elements are equivalence classes of measurable functions. ( $f \sim g$ if and only if $d(f, g)=0$.) Assuming that ( $X, \mathfrak {M}, \mu$ ) is a finite measure space, this is indeed a metric space. In Exercises 7.2.35-7.2.38, we will explore some of its properties.

Prove that the function

defines another metric in the space $M(X, \mu )$ and that convergence in the space $M(X, \mu )$ with respect to $\rho$ is equivalent to the convergence in measure.

Let $(X, \mathfrak {M}, \mu )$ be a finite measure space and, as in Exercise 7.2.34, for measurable functions $f, g$ on $X$ let $d(f, g)=\int_{X} \frac{|f-g|}{1+|f-g|} d \mu$. Exercise 7.2.34 motivates the construction of a metric space $M(X, \mu )$ whose elements are equivalence classes of measurable functions. ( $f \sim g$ if and only if $d(f, g)=0$.) Assuming that ( $X, \mathfrak {M}, \mu$ ) is a finite measure space, this is indeed a metric space. In Exercises 7.2.35-7.2.38, we will explore some of its properties.

Prove that the only continuous linear map $F: M(X, \mu ) \rightarrow \mathbb {R}$ is the zero map.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space. Prove or disprove: a sequence of real-valued measurable functions $\left\{ f_{n}\right\}$ converges in measure to $f$ if and only if

Suppose that every point in $[0,1]$ belongs to at least $k$ sets of the family of Lebesgue measurable sets $\left\{ E_{k}\right\}_{k=1}^{N}$. Prove that for some $n \in \left\{ 1,2, \ldots , N\right\}$, $m\left(E_{n}\right) \geq k / N$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space. Prove that $X$ is $\sigma$-finite if and only if there is a strictly positive, real-valued function $f \in L(X, \mu )$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space and let $\left\{ A_{n}\right\} \subset \mathfrak {M}$ such that $\sum_{n=1}^{\infty } \mu \left(A_{n}\right)<\infty$. For each $m \in \mathbb {N}$ let $B_{m}$ consist of those points of $X$ that belong to $A_{n}$ for at least $m$ distinct values of $n$. Prove that each $B_{m}$ is measurable and that $\mu \left(B_{m}\right) \leq (1 / m) \sum_{n=1}^{\infty } \mu \left(A_{n}\right)$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space and $\left\{ f_{n}\right\}$ a sequence of measurable functions on $X$. Let $g$ be a nonnegative function that is integrable over $X$ and $\left|f_{n}\right| \leq g$ a.e on $X$, for all $n \in \mathbb {N}$. Prove that each $f_{n}$ is integrable over $X$ and so are the limit inferior and the limit superior of $\left\{ f_{n}\right\}$.

Let ( $X, \mathfrak {M}, \mu$ ) be a finite measure space and let $f \in L(X, \mu )$. Let $P= \left\{ \ldots , x_{-2}, x_{-1}, x_{0}, x_{1}, x_{2}, \ldots \right\}$ be a partition of $\mathbb {R}$, and let $\xi_{k} \in \left[x_{k}, x_{k+1}\right]$. Prove that

Prove that the assertion in Exercise 7.3.2 remains true if $\mu (X)=\infty$, provided that $\xi_{k}=0$ for all $k$ for which $0 \in \left[x_{k}, x_{k+1}\right]$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space and let $f$ be a bounded measurable function on $X$. Suppose that there are constants $A>0$ and $\alpha <1$ such that, for every $\varepsilon >0$,

Prove that $f$ is integrable over $X$.

Let ( $X, \mathfrak {M}, \mu$ ) be a finite measure space and let $f$ be a measurable function on $X$. Suppose that there are constants $A>0$ and $\alpha >1$ such that, for every $M>0$,

Prove that $f$ is integrable over $X$.

Suppose that a sequence of measures $\left\{ \mu_{i}\right\}$ on a measurable space ( $X, \mathfrak {M}$ ) converges to a measure $\mu$, meaning that for every $A \in \mathfrak {M}, \lim \mu_{n}(A)=\mu (A)$. Prove or disprove: if $f$ is integrable with respect to each $\mu_{n}$ then it is integrable with respect to $\mu$.

Let $(X, \mathfrak {M}, \mu )$ be a $\sigma$-finite measure space, let $f: X \rightarrow \mathbb {R}$ be measurable and suppose that there is $C>0$ such that for every set $E$ of finite measure, $\left|\int_{E} f d \mu \right| \leq C$. Prove that $f$ is integrable over $X$.

Let $(\mathbb {R}, \mathfrak {M}, \mu )$ be a finite measure space, and $f: \mathbb {R} \rightarrow \mathbb {R}^{*}$ a measurable function. Prove that $f \in L(\mathbb {R}, \mu )$ if and only if

Prove that in Egorov's Theorem the hypothesis that $(X, \mathfrak {M}, \mu )$ is a finite measure space can be replaced by $\left|f_{n}\right| \leq g$ for all $n \in \mathbb {N}$ where $g$ is integrable over $X$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space, let $f: X \rightarrow \mathbb {R}$ be measurable, and let $\left\{ X_{n}\right\}$ be disjoint measurable sets such that $X=\cup_{n=1}^{\infty } X_{n}$. Characterize the integrability of $f$ over $X$ in terms of the integrability of $f$ over $X_{n}$ and $\int_{X_{n}} f d \mu$.

Let $(X, \mathfrak {M}, \mu )$ be a measure space and $\left\{ f_{n}\right\}$ a sequence of measurable functions on $X$. Let $g$ be a nonnegative function that is integrable over $X$ and $\left|f_{n}\right| \leq g$ a.e on $X$, for all $n \in \mathbb {N}$. Prove that

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space and $\left\{ f_{n}\right\}$ a sequence of measurable functions on $X$ that converges in measure to a measurable function $f$. Let $g$ be a function that is integrable over $X$ such that $f_{n} \geq -g$ for all $n$. Prove that

Let ( $X, \mathfrak {M}, \mu$ ) be a finite measure space and let $\left\{ f_{n}\right\}$ be a sequence of integrable function over $X$ that converges uniformly to a function $f$. Prove that

What if $\mu (X)=\infty$ ?

Let $f:[a, b] \rightarrow \mathbb {R}$ be a Lebesgue integrable function. Prove that there exists a constant $C_{0}$ such that

Prove that such $C_{0}$ is unique if the function $f$ is continuous.

Let $f:[0,+\infty ) \rightarrow \mathbb {R}$ be a continuous function ans suppose that $\lim_{x \rightarrow +\infty } f(x)=\gamma$. Prove that for any $a>0$,

Let $(X, \mathfrak {M}, \mu )$ be a measure space, let $\left\{ f_{n}\right\} ,\left\{ g_{n}\right\} , f, g$ be integrable functions over $X$ such that $\lim f_{n}=f$ and $\lim g_{n}=g$ a.e. on $X$. Suppose that $\left|f_{n}\right| \leq g_{n}$ a.e on $X$, for all $n \in \mathbb {N}$, and that $\lim \int_{X} g_{n} d \mu =\int_{X} g d \mu$. Prove that

Let $\left\{ f_{n}\right\}$ be a sequence of integrable functions on $[0,1]$, let $f$ be an integrable function on $[0,1]$, and suppose that $f_{n} \rightarrow f$ a.e. on $[0,1]$. Prove that $\lim_{n \rightarrow \infty } \int_{[0,1]}\left|f-f_{n}\right|=0$ if and only if $\lim_{n \rightarrow \infty } \int_{[0,1]}\left|f_{n}\right|=\int_{[0,1]}|f|$.

Let $\left\{ f_{n}\right\}$ be a sequence of Lebesgue measurable functions on $[0,1]$ that converges to a function $f$ a.e. and suppose that there is $M>0$ such that for all $n \in \mathbb {N},\left|f_{n}\right| \leq M$ a.e. Prove that the sequence

converges uniformly to 0 on $[0,1]$.

Let $f$ be an integrable function on a Lebesgue measurable set $E \subset \mathbb {R}^{n}$.

Suppose that $f$ is integrable over $[0,1]$. Calculate

Suppose that $f$ is integrable over $[0,1]$ and that $\lim_{x \rightarrow 1-} f(x)=A$. Prove that

Let $X=[0,1], \mathfrak {M}=\mathcal{M} \mid [0,1]$, and $\mu =m \mid [0,1]$. Let $f$ be integrable over $X$. Find

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space, let $\left\{ f_{n}\right\}$ be an increasing sequence of integrable functions, and suppose that $\sup_{n} \int_{X} f_{n} d \mu <\infty$. Prove that $f:= \lim_{n} f_{n}$ is integrable and $\lim_{n} \int_{X} f_{n} d \mu =\int_{X} f d \mu$.

Let $\phi : \mathbb {R} \rightarrow \mathbb {R}$ be a bounded function in $C^{1}(\mathbb {R})$ with a bounded derivative, let $\phi (0)=0$, and let $g$ be a function that is integrable over $\mathbb {R}$. Prove that

is well-defined and differentiable for all $t \in \mathbb {R}$.

Let $f_{n}:[0,1] \rightarrow [0, \infty )$ be Lebesgue integrable for $n=1,2, \ldots$. Suppose that for every $n \in \mathbb {N}$,

Finally, let $g(x)=\sup \left\{ f_{n}(x): n \in \mathbb {N}\right\}$. Prove that $\int_{0}^{1} g=\infty$.

Let $\left[a_{i j}\right]$ be an infinite matrix of real numbers such that $\left|a_{i j}\right| \leq 1$ for all $i, j \in \mathbb {N}$. Suppose that the limit along each row is 0 , i.e., $\lim_{j \rightarrow \infty } a_{i j}=0$ for all $i \in \mathbb {N}$. If $p>1$ prove that $\lim_{j \rightarrow \infty } \sum_{i=1}^{\infty } a_{i j} / i^{p}=0$.

Let $(X, \mathfrak {M}, \mu )$ be a measure space, let $\left\{ f_{n}\right\} , f$ be nonnegative integrable functions over $X$ such that $\lim f_{n}=f$ a.e. on $X$ and $\lim \int_{X} f_{n} d \mu =\int_{X} f d \mu$. Prove that

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space, let $f \in L(X, \mu )$, and let $A_{n}=\left\{ x \in X: 1 / n \leq |f(x)| \leq n\right\}$, for $n \in \mathbb {N}$. Prove that

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space, let $\left\{ f_{n}\right\}$ be integrable functions over $X$ such that $\sum \int_{X}\left|f_{n}\right|<\infty$. Prove that $\sum_{1}^{\infty } f_{n}$ converges a.e. to an integrable function and $\int_{X} \sum_{1}^{\infty } f_{n} d \mu =\sum_{1}^{\infty } \int_{X} f_{n} d \mu$.

Let $f:[0,1] \times [0,1] \rightarrow \mathbb {R}$ be a measurable function of $x$, a continuous the function of $y$, and let $g:[0,1] \rightarrow \mathbb {R}$ be integrable over $[0,1]$ satisfying $|f(x, y)| \leq g(x)$ for $(x, y) \in [0,1] \times [0,1]$. Prove that $h(y):=\int_{0}^{1} f(x, y) d x$ is continuous.

Prove the Dominated Convergence Theorem under a weaker assumption that $\lim f_{n}=f$ in measure (instead of a.e.).

Let $f: \mathbb {R} \rightarrow \mathbb {R}^{*}$ be a continuous function, let $a \in \mathbb {R}$, and let

Prove that

Let $\delta_{0}$ be the Dirac measure on $\mathbb {R}$ concentrated at $x=0$ and let $f$ be a function on $\mathbb {R}$. Calculate

Let $f$ be a Lebesgue integrable function on $\mathbb {R}$ and suppose that $\int_{E} f=$ 0 for every measurable set $E \subset \mathbb {R}$ with $m(E)=\pi$. Prove or disprove: $f=0$ a.e.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space, let $f: X \rightarrow \mathbb {R}$ be measurable, and let $F$ be a closed subset of $\mathbb {R}$ such that for every measurable set $E$ of positive measure