I'm currently trying to understand PPPs. In the following I will state what I believe to know (please correct me if I'm wrong). I'm considering a PPP with intensity $\lambda$ on area $A = [-0.5, 0.5] \times [-0.5,0.5]$. Intensity means that there are $\lambda$ points per unit area and, therefore, here in total $\lambda$ points in $A$.
Now, the probability of having $k$ points in $B \subseteq A$ is given by:
$p( N(B) =k ) = \frac{(\lambda l(B))^k}{k!} \exp{(- \lambda l(B))}$.
where $l(B)$ denotes the Lebesgue measure of $B$.
From my understanding, I would have assumed that the probability of having $\lambda$ points in $A$ is exactly one.
$p( N(A)=\lambda) = \frac{(\lambda)^k}{\lambda!} \exp{(- \lambda)} \overset{!}{=}1 $
because $l(A)=1$. But this doesn't seem to be the case, why?
Because $N(A)$ is random and it may well happen that some random variables $X$ are such that $X=E(X)$ never happens. In your case, if $\lambda$ is not an integer, you see that $N(A)=\lambda$ never happens since $N(A)$ is integer valued (and, even when $\lambda$ is an integer, $P(N(A)=\lambda)<1$, not $=1$).
To sum up: do not confuse random variables (functions) with their expectations (numbers).