The Lebesgue Integral

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: