  We consider the critical FK-Ising measure $\phi_{\beta_c}$ on $\mathbb Z^d$
with $d\geq 3$. We construct the measure $\phi^\infty:=\lim_{|x|\rightarrow
\infty}\phi_{\beta_c}[\:\cdot\: |\: 0\leftrightarrow x]$ and prove it satisfies
$\phi^\infty[0\leftrightarrow \infty]=1$. This corresponds to the natural
candidate for the incipient infinite cluster measure of the FK-Ising model. Our
proof uses a result of Lupu and Werner (Electron. Commun. Probab., 2016) that
relates the FK-Ising model to the random current representation of the Ising
model, together with a mixing property of random currents recently established
by Aizenman and Duminil-Copin (Ann. Math., 2021). We then study the
susceptibility $\chi(\beta)$ of the nearest-neighbour Ising model on $\mathbb
Z^d$. When $d>4$, we improve a previous result of Aizenman (Comm. Math. Phys.,
1982) to obtain the existence of $A>0$ such that, for $\beta<\beta_c$,
\begin{equation*}
  \chi(\beta)= \frac{A}{1-\beta/\beta_c}(1+o(1)), \end{equation*} where $o(1)$
tends to $0$ as $\beta$ tends to $\beta_c$. Additionally, we relate the
constant $A$ to the incipient infinite cluster of the double random current.
