Integration

Prove that if $f$ and $g$ are integrable simple functions, then

Suppose that $\mu \left(X\right)<\infty$ and that $f$ is an integrable function on $X$. Prove that the Lebesgue integral $\int_X f(x)\, d\mu$ can be computed by the formula

where $T=\left\{ t_k\right\}$ is a partition of the real axis, $\lambda (T)=\sup_k\left|t_k-t_{k+1}\right|$ is the diameter of the partition $T$, and $\left\{ \xi_k\right\}$ is any set of points satisfying the condition $\xi_k\in [t_k,t_{k+1}]$. The expression in (1) is called a Lebesgue sum.

Prove that the statement of Problem 163 remains true in the case $\mu \left(X\right)=\infty$ if it is required in addition that $\xi_k=0$ for those $k$ such that $[t_k,t_{k+1}]$ contains the point 0.

Suppose that the measurable simple function $f$ is represented in two ways as a linear combination of characteristic functions of disjoint sets:

Prove that $\sum_k c_k\mu \left(A_k\right)=\sum_l d_l\mu \left(B_l\right)$ in the case when one of these series converges absolutely.

Let $f_n$ be the simple function on $[0,1]$ defined by the formula $f_n(x)=(1/n)\left[nx\right]$, where $\left[x\right]$ denotes the integral part of the number $x$. Prove that the sequence $\left\{ f_n\right\}$ is Cauchy, but does not have a limit in the space $S[0,1]$ of simple integrable functions with the distance $d_1(f,g)=\int_0^1\left|f-g\right|\, dx$.

For what values of the parameters $\alpha$ and $\beta$ is the function $f(x)=x^\alpha \sin x^\beta$ on $(0,1]$

Prove that the integral of a nonnegative integrable function $f$ over a set $A$ is:

Let $\varphi$ be a monotonically increasing smooth function on $[a,b]$, and $\psi$ the inverse of this function on $[\varphi (a),\varphi (b)]$. Regarding the integral as a limit of Lebesgue sums, prove the identity

Prove that the Lebesgue integral of a nonnegative function $f(x)$ over $[a,b]$ coincides with the Lebesgue measure of the planar set determined by the inequalities $a\leq x\leq b$, $0\leq y\leq f(x)$.

For any real function $f$ let $f_+(x)=(f(x)+\left|f(x)\right|)/2$, $f_-(x)=(\left|f(x)\right|-f(x))/2$. Prove that $f$ is integrable if and only if $f_+$ and $f_-$ are integrable.

Prove that a measurable nonnegative function $f$ is integrable if and only if $\sup \int_A f(x)\, d\mu (x)<\infty$, where the supremum runs over all sets $A$ of finite measure on which $f$ is bounded above.

Let $\mu \left(X\right)<\infty$. Prove that a nonnegative measurable function $f$ on $X$ is integrable if and only if the series

converges.

Prove that a nonnegative bounded function on a set $X$ of infinite measure is integrable if and only if the series

converges.

Prove that a bounded function is Riemann integrable on $[a,b]$ if and only if it is almost everywhere continuous.

Prove that the Lebesgue integral of the function

is finite if and only if the symmetric matrix $A=\left\| a_{ij}\right\|$ is positive-definite. Prove that in this case the integral is equal to $\operatorname {det}\left(\pi \cdot A^{-1}\right)$.

Compute the integral with respect to Wiener measure on $C[0,1]$ of the function

Let $C_0[0,1]$ be the space of continuous functions $x(t)$ on $[0,1]$ satisfying the additional condition $x(0)=0$. Prove that $C[0,1]$ can be identified with the product $\mathbb {R}\times C_0[0,1]$, and that the Wiener measure $\mu$ passes into $\mu_1\times \mu_0$, where $\mu_1$ is the usual Lebesgue measure on $\mathbb {R}$ and $\mu_0$ is a certain measure on $C_0[0,1]$.

Let $\mu_0$ be the measure constructed in Problem 179. Compute the integrals:

Establish the following properties of the total variation:

Prove that the set of points of discontinuity of a function of bounded variation on a segment is not more than countable and consists only of points of discontinuity of the first kind.

Prove that a function of bounded variation on a closed interval is Lebesgue measurable.

Prove that a function with a bounded derivative on a closed interval is of bounded variation.

Suppose that $f$ has a Riemann integrable derivative on $[a,b]$. Prove the formula

Let $\Phi$ be a left-continuous function of bounded variation on $[a,b]$. Prove that $\Phi$ has a unique representation as a sum $\Phi =\Phi_0+\Phi_1$, where $\Phi_0$ is a continuous function of bounded variation, and $\Phi_1$ is a so-called jump function: $\Phi_1(x)=\sum_{k=1}^{\infty }c_k\theta (x-a_k)$, where $\left\{ a_k\right\}$ is a finite or countable subset of $[a,b]$, $\theta (x)$ is the Heaviside function, defined by

and $\left\{ c_k\right\}$ is a numerical sequence satisfying the condition $\sum_k\left|c_k\right|<\infty$.

Prove that:

Determine whether the function $\varphi (f)$ has bounded variation on $[0,1]$ if $f$ has bounded variation on $[0,1]$ and $\varphi$

Let $E$ be a subset of $[0,1]$, and $\chi$ the characteristic function of $E$. Prove that $\chi$ has bounded variation if and only if the boundary of $E$ is a finite set.

Let $f$ and $g$ be two continuous functions with bounded variation on $[a,b]$. Prove that the set $(f(x),g(x))$, $x\in [a,b]$, cannot fill the square. Is this true without the requirement of bounded variation?

Prove the following properties of the Riemann–Stieltjes integral:

Suppose that the function $\Phi$ has bounded variation on $[a,b]$ and is discontinuous at $c\in (a,b)$, while the function $f$ is Riemann–Stieltjes integrable with respect to $\Phi$. Prove that $f$ is continuous at the point $c$.

Prove that if $\Phi$ is a function of bounded variation on $[a,b]$ that differs from zero only at a finite or countable number of points lying in $(a,b)$, then for any function $f$ continuous on $[a,b]$,

Prove that if $f$ is continuous, then the Riemann–Stieltjes integral $\int_a^b f(x)\, d\Phi (x)$ does not depend on the values taken by $\Phi$ at points of discontinuity lying in $(a,b)$.

Prove the formula for integration by parts for the Stieltjes integral:

Suppose that $f(x)$ is continuous on $[a,b]$ and that $g(x)$ has a Riemann integrable derivative $g'(x)$ defined everywhere except at finitely many points $c_1,\ldots ,c_k$. Prove that under these conditions the Riemann–Stieltjes integral $\int_a^b f\, dg$ exists and that it can be expressed by the formula

Let $\mu_\varphi$ be the measure generated by a monotone continuous function $\varphi$. Prove that the Lebesgue integral $\int_{[a,b]}x\, d\mu_\varphi$ is equal to the Stieltjes integral $\int_a^b x\, d\varphi (x)$, and compute it.

Let $\varphi (x)$ be the Cantor staircase, i.e., the continuous monotone function on $[0,1]$ constant on each interval in the complement of the Cantor set and taking the values $1/2^k$, $3/2^k$, $5/2^k$, $\ldots$, $(2^k-1)/2^k$ on the intervals of rank $k$.

Compute the integrals:

Suppose that the sequence $f_n\in L^{1}\left(X,\mu \right)$ converges uniformly to the function $f(x)$. Prove that $f_n\to f$ in $L^{1}\left(X,\mu \right)$ if $\mu \left(X\right)<\infty$. Is this true in the case $\mu \left(X\right)=\infty$?

Let $X$ be a set of finite $\mu$ measure. For any measurable functions $f$ and $g$ let

Let $\left\{ f_n\right\}$ be a sequence of nonnegative integrable functions converging almost everywhere to an integrable function $f$. Prove that if $\int_X f_n\, d\mu \to \int_X f\, d\mu$ as $n\to \infty$, then $f_n\to f$ in the space $L^{1}\left(X,\mu \right)$.

Suppose that $f\in L^{1}\left(X,\mu \right)$ and $\mu \left(X\right)=1$. Prove that there is a monotone function $g(t)\in L^{1}\left(\left[0,1\right]\right)$ such that for any $t\in [0,1]$,

Let $\mu$ be a nonzero Borel measure on the set of real numbers that has the following property: for any $t\in \mathbb {R}$ the measure $\mu_t$ defined by the formula $\mu_t\left(A\right)=\mu \left(A+t\right)$ is equivalent to $\mu$. (Such measures are said to be quasi-invariant under translations.) Prove that $\mu$ is equivalent to Lebesgue measure.

Let $\mu$ be a measure on $X$, and $f_1,f_2$ two $\mu$-integrable real functions on $X$. Define signed measures $\nu_i=f_i\mu$ by the formula $\nu_i\left(A\right)=\int_A f_i\, d\mu$, $i=1,2$. Prove that $\nu_1$ and $\nu_2$ are equivalent if and only if $\mu \left(N_1\triangle N_2\right)=0$, where $N_i=\left\{ x\in X:f_i(x)\neq 0\right\}$.

Let $\mu$ be a $\sigma$-finite measure on $X$, and $\nu$ a measure defined on the same $\sigma$-algebra and absolutely continuous with respect to $\mu$ (i.e., $\mu \left(A\right)=0\Rightarrow \nu \left(A\right)=0$). Prove that there exists a nonnegative $\mu$-measurable function $\rho$ having the property that $\nu \left(A\right)=\int_A\rho (x)\, d\mu (x)$ for any measurable set $A$ (both sides of the equation may simultaneously take the value $+\infty$).

Prove that there does not exist a Lebesgue-measurable subset $A$ of the real line $\mathbb {R}$ such that for any interval $\Delta$

Let f\in L^{1}\left({(a,b\right)}]. Prove that the function $F(x)=\int_a^x f(x)\, d\mu (x)$ is differentiable almost everywhere and $F'(x)=f(x)$ for almost all $x\in [a,b]$.

A real function $F$ on $[a,b]$ is said to be absolutely continuous if for any $\varepsilon >0$ there is a $\delta >0$ such that $\sum_{i=1}^n\left|F(a_i)-F(b_i)\right|<\varepsilon$ for any family of intervals $\left\{ \Delta_i\right\}$, $\Delta_i=(a_i,b_i)$, $1\leq i\leq n$, with sum of lengths $<\delta$. Prove that:

Prove that the following sets are dense in $L^{1}\left(\left[0,1\right]\right)$:

Prove that the following sets are dense in $L^{1}\left(\mathbb {R}\right)$:

Let $f\in L^{1}\left(\mathbb {R}\right)$. Prove that $\int_{\mathbb {R}}\left|f(x+\varepsilon )-f(x)\right|\, dx\to 0$ as $\varepsilon \to 0$. In other words, translation is a continuous operation in $L^{1}\left(\mathbb {R}\right)$.

The convolution of functions $f_1$ and $f_2$ on the line is defined to be the function $f$ given by the formula