I thought that it might be instructive to present a way forward that complements the solutions that have already been posted. To that end, we proceed.

Let $u$ be the unit step function and $\phi\in \mathbb{S}$. Then, the Fourier transform of the unit step, $\mathscr{F}\{u\}$, is defined by the functional

$$\begin{align}
\langle \mathscr{F}\{u\}, \phi\rangle &=\langle u,\mathscr{F}\{\phi\},\rangle\\\\
&=\int_0^\infty \int_{-\infty}^\infty \phi(t)e^{-i\omega t}\,dt\,d\omega\\\\
&=\lim_{L\to\infty}\int_{-\infty}^\infty \phi(t) \,\frac{1-e^{-itL}}{it}\,dt\tag1
\end{align}$$
where the Fubini-Tonelli theorem justifies the interchange of integrals that resulted in $(1)$.

For any $\delta>0$, we write the right-hand side of $(1)$ as
$$\begin{align}
\langle \mathscr{F}\{u\}, \phi\rangle&=\lim_{L\to\infty}\left(\int_{|t|\le \delta} \phi(t) \,\frac{1-e^{-itL}}{it}\,dt+\int_{|t|\ge \delta} \phi(t) \,\frac{1-e^{-itL}}{it}\,dt\right)\\\\
&=\lim_{L\to\infty}\left(\int_{|t|\le \delta} \phi(t) \,\frac{1-e^{-itL}}{it}\,dt\right)+\int_{|t|\ge \delta} \frac{\phi(t)}{it}\,dt\tag2\\\\
\end{align}$$

We applied the Riemann-Lebesgue Lemma to arrive at $(2)$.

Next, we write the first integral on the right-hand side of $(2)$ as
$$\begin{align}
\int_{|t|\le \delta} \phi(t) \,\frac{1-e^{-itL}}{it}\,dt&=\phi(0)\int_{|t|\le \delta} \frac{1-e^{-itL}}{it}\,dt+\int_{|t|\le \delta} (\phi(t)-\phi(0)) \,\frac{1-e^{-itL}}{it}\,dt\\\\
&=\phi(0)\int_{|t|\le \delta} \,\frac{\sin(Lt)}{t}\,dt+O(\delta)\tag3
\end{align}$$

where we exploited the odd symmetry of $\frac{1-\cos(Lt)}{it}$ and the smoothness of $\phi$ to arrive at $(3)$.

Letting $L\to \infty$ in $(3)$, and substituting into $(2)$, reveals $$\begin{align}
\langle \mathscr{F}\{u\}, \phi\rangle&=\pi \phi(0)+O(\delta)+\int_{|t|\ge \delta} \frac{\phi(t)}{it}\,dt\tag 4
\end{align}$$

Finally, letting $\delta\to0^+$ yields the coveted result
$$\langle \mathscr{F}\{u\}, \phi\rangle=\pi \phi(0)+\text{PV}\int_{-\infty}^\infty \frac{\phi(t)}{it}\,dt$$

from which we deduce that
$$\bbox[5px,border:2px solid #C0A000]{\mathscr{F}\{u\}(\omega)=\pi \delta(\omega)+\text{PV}\left(\frac1{i\omega}\right)}$$