Product Measures, Convolutions, and Transforms

Show that $D_{1} \cup D_{2}$ is countable.

Let $\phi_{i}(x) = \mu_{i}(\left\{ x\right\} ) x \in \mathbb {R}, i = 1,2$. Show that $\phi_{i}$ is Borel measurable for $i = 1,2$.

Show that $\int \phi_{1} d \mu_{2} = \sum_{z \in D_{1} \cap D_{2}} \phi_{1}(z) \phi_{2}(z)$.

Deduce (2.11) from (c).

Show that for any $h: \mathbb {R}_{+} \rightarrow \mathbb {R}_{+}$that is absolutely continuous,

Show that for any $0 < p < \infty$,

Show that for any $0 < p < \infty$,

where $\Gamma (p) = \int_{[0, \infty )} e^{-t} t^{p-1} d t, p > 0$, and $\psi_{X}(t) = \int_{\Omega } e^{-t X} d P$, $t \in \mathbb {R}_{+}$. (Hint: (a) Apply Tonelli's theorem to the function $f(t, \omega ) \equiv h^{\prime }(t) I(X(w) \geq t)$ on the product measure space $([0, \infty ) \times \Omega , \mathcal{B}([0, \infty )) \times \mathcal{F}, m \times P)$, where $m$ is Lebesgue measure on $\left(\mathbb {R}_{+}, \mathcal{B}\left(\mathbb {R}_{+}\right)\right).$)

Show that $A \in \mathcal{B}\left(\mathbb {R}^{2}\right)$.

Show that

where $m^{(2)}$ is Lebesgue measure on $\left(\mathbb {R}^{2}, \mathcal{B}\left(\mathbb {R}^{2}\right)\right)$ and $m(\cdot )$ is Lebesgue measure on $\mathbb {R}$.

If $g$ is continuous and strictly increasing show that the two integrals in (b) equal

For $1 < A < \infty$, let

Use integration by parts to show that

and

Show using Fubini's theorem that for $0 < A < \infty$,

Conclude using the DCT that

Using Theorem 4.4.1 and the fact that $\phi (x) \equiv \tan x$ is a (1-1) strictly monotone map from $\left(0, \frac{\pi }{2}\right)$ to $(0, \infty )$ having the inverse $\operatorname {map} \psi (\cdot )$ with derivative $\psi^{\prime }(t) \equiv \frac{1}{1+t^{2}}, 0 < t < \infty$, conclude that

Show that for any Borel measurable $f: \mathbb {R} \rightarrow \mathbb {R}_{+}, f(x+y)$ is $\langle \mathcal{B}(\mathbb {R}) \times \mathcal{B}(\mathbb {R}), \mathcal{B}(\mathbb {R})\rangle$-measurable from $\mathbb {R} \times \mathbb {R} \rightarrow \mathbb {R}$ and

Show that

Suppose there exist countable sets $B_{\lambda }, B_{\mu }$ such that $\mu \left(B_{\mu }^{c}\right) = 0 = \lambda \left(B_{\lambda }^{c}\right)$. Show that there exists a countable set $B_{\nu }$ such that $\nu \left(B_{\nu }^{c}\right) = 0$.

Suppose that $\mu (\left\{ x\right\} ) = 0$ for all $x$ in $\mathbb {R}$. Show that $\nu (\left\{ x\right\} ) = 0$ for all $x$ in $\mathbb {R}$.

Suppose that $\mu \ll m$ with $\frac{d \mu }{d m} = h$. Show that $\nu \ll m$ and find $\frac{d \nu }{d m}$ in terms of $h, \mu$ and $\lambda$.

Suppose that $\mu \ll m$ and $\lambda \ll m$. Show that

Show that $\left(F_{1} * F_{2}\right)(x) \equiv \int_{\mathbb {R}} F_{1}(x-u) d F_{2}(u)$ is well defined and is a cdf on $\mathbb {R}$.

Show also that $\left(F_{1} * F_{2}\right)(\cdot ) = \left(F_{2} * F_{1}\right)(\cdot )$.

Suppose $t \in \mathbb {R}$ is such that $\int e^{t x} d F_{i}(x) < \infty$ for $i = 1,2$. Show that $\int e^{t x} d\left(F_{1} * F_{2}\right)(x) = \left(\int e^{t x} d F_{1}(x)\right)\left(\int e^{t x} d F_{2}(x)\right)$.

Show that if $f$ is continuous and bounded on $\mathbb {R}$, then so is $f * g$.

Show that if $f$ is differentiable with a bounded derivative on $\mathbb {R}$, then so is $f * g$. (Hint: Use the DCT.)

Show that if $1 \leq p < \infty$, then for all $x$ in $\mathbb {R}$

and hence that

is well defined a.e. $(m)$ and

with " = " holding iff either $f = 0$ a.e. or $g = 0$ a.e.

Show that if $p = \infty$ then

and " $=$ " can hold for some nonzero $f$ and $g$. (Hint for (a): Use Jensen's inequality with probability measure $d \mu = \frac{\left|f\right| d m}{\left\| f\right\|_{1}}$ if $\left\| f\right\|_{1} > 0$.)

Show that $f * g$ is well defined and uniformly continuous.

Show that if $1 < p < \infty$,

(Hint: For (a) use Hölder's inequality and Lemma 5.7.7. For (b) approximate $g$ by simple functions.)

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be Borel measurable and $\int_{A}\left|f\right| d m < \infty$ for all bounded intervals $A$ in $\mathbb {R}$. Show that $f * g$ is well defined and infinitely differentiable.

Show that for any $f \in L^{1}(\mathbb {R})$, there exist a sequence $\left\{ g_{n}\right\}_{n \geq 1}$ of such functions such that $f * g_{n} \rightarrow f$ in $L^{1}(\mathbb {R})$.

Show that $f_{\sigma }$ is infinitely differentiable.

Show that if $f$ is continuous and zero outside a bounded interval, then $f_{\sigma }$ converges to $g$ uniformly on $\mathbb {R}$ as $\sigma \rightarrow 0$.

Show that if $f \in L^{p}(\mathbb {R}), 1 \leq p < \infty$, then $f_{\sigma } \rightarrow f$ in $L^{p}(\mathbb {R})$ as $\sigma \rightarrow 0$.

Show that $h = f * g$ for some $g$ bounded and with bounded support.

Show that $h(\cdot )$ is continuous and that $\lim_{\left|x\right| \rightarrow \infty } h(x) = 0$. (Hint: For $1 < p < \infty$, use Hölder's inequality and show that $m(\left(A+x_{1}\right) \triangle \left(A+x_{2}\right)) \rightarrow 0$ as $\left|x_{1}-x_{2}\right| \rightarrow 0$.)

Verify the claims in Examples 5.4.1 directly.

Verify the same using generating functions.

Let $f \in L^{1}(\mathbb {R})$ and $x^{k} f(x) \in L^{1}(\mathbb {R})$ for some $k \geq 1$. Show that $\hat{f}(\cdot )$ is $k$-times differentiable on $\mathbb {R}$ with all derivatives $\hat{f}^{(r)}(t) \rightarrow 0$ as $\left|t\right| \rightarrow \infty$ for $r \leq k$.

Let $f \in L^{1}(\mathbb {R})$. Suppose $\int \left|t \hat{f}(t)\right| d t < \infty$. Show that there exists a function $g: \mathbb {R} \rightarrow \mathbb {R}$ such that it is differentiable, $\lim_{\left|x\right| \rightarrow \infty }\left(\left|g(x)\right|+\left|g^{\prime }(x)\right|\right) = 0$ and $g = f$ a.e. $(m)$. Extend this to the case where $\int \left|t^{k} \hat{f}(t)\right| d t < \infty$ for some $k > 1$. (Hint: Consider $g(x) = \frac{1}{2 \pi } \int e^{-\iota t x} \hat{f}(t) d t$.)

Show that $\hat{f}(\cdot )$ is real valued, differentiable and satisfies the ordinary differential equation $\hat{f}^{\prime }(t)+t \hat{f}(t) = 0, t \in \mathbb {R}$ and $\hat{f}(0) = 1$. Find $\hat{f}(t)$.

For $\mu$ in $\mathbb {R}, \sigma > 0$, let

Find $\hat{f}_{\mu , \sigma }$ and verify that for any $\left(\mu_{i}, \sigma_{i}\right), i = 1,2$,

(Hint for (b): Use Fourier transforms and uniqueness.)

Find $\hat{h}(n) \equiv \frac{1}{2 \pi } \int_{-\pi }^{\pi } e^{-\iota n x} h(x) d x, n = 0, \pm 1, \pm 2$.

Show that $\sum_{n = -\infty }^{\infty }\left|\hat{h}(n)\right| < \infty$.

Show that $S_{n}(h, x) \equiv \sum_{j = -n}^{+n} \hat{h}(j) e^{\iota j x}$ converges to $h(x)$ uniformly on $[-\pi , \pi ]$.

Verify that

and

(Remark: This example shows that the Fourier series of a function can converge very slowly such as in this example where the rate of decay is $\frac{1}{n}$.)