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4

$$\mathbf 1_{(x,x+a]}(y)=1\iff x\lt y\leqslant x+a\iff y-a\leqslant x\lt y\iff \mathbf 1_{[y-a,y)}(x)=1$$

2

We have \begin{align} \frac{e^{-ita}-e^{-itb}}{it}e^{itx} &= \frac{e^{it(x-a)}-e^{it(x-b)}}{it}\\ &= \frac{\cos\left( t(x-a)\right) - \cos \left(t(x-b)\right)}{it} + \frac{\sin \left(t(x-a)\right) - \sin \left(t(x-b)\right)}{t} \end{align} by the addition theorem for the exponential function and Euler's formula $e^{iz} = \cos z + i\sin z$. The ...

1

Let $O \subseteq \mathbb{R}$ be an open set and define $$O_n := \left\{x \in \mathbb{R}; d(x,\mathbb{R} \backslash O) \geq \frac{1}{n} \right\}.$$ It is not difficult to show that $O_n$ is closed and $O_n \uparrow O$. Now choose $n$ sufficiently large such that $m(O \backslash O_n)<\varepsilon$. It is not difficult to show that there exists a continuous ...

1

Having done what you've done up to the left side of the last line, now change variables to $u=z-w$ and do essentially the same thing again. You can also think of this without integrals: what is the measure of $(-1/2,1/2) \cap \{ w : z - w \in (-1/2,1/2) \}$? Well, the second set is $(z-1/2,z+1/2)$, so what happens? (As your work suggests, it depends on ...

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