  Let $A$ be an $n\times n$ real expanding matrix and $\mathcal{D}$ be a finite
subset of $\mathbb{R}^n$ with $0\in\mathcal{D}$. The family of maps
$\{f_d(x)=A^{-1}(x+d)\}_{d\in\mathcal{D}}$ is called a self-affine iterated
function system (self-affine IFS). The self-affine set $K=K(A,\mathcal{D})$ is
the unique compact set determined by $(A, {\mathcal D})$ satisfying the
set-valued equation $K=\displaystyle\bigcup_{d\in\mathcal{D}}f_d(K)$. The
number $s=n\,\ln(\# \mathcal{D})/\ln(q)$ with $q=|\det(A)|$, is the so-called
pseudo similarity dimension of $K$. As shown by He and Lau, one can associate
with $A$ and any number $s\ge 0$ a natural pseudo Hausdorff measure denoted by
$\mathcal{H}_w^s.$ In this paper, we show that, if $s$ is chosen to be the
pseudo similarity dimension of $K$, then the condition $\mathcal{H}_w^s(K)> 0$
holds if and only if the IFS $\{f_d\}_{d\in\mathcal{D}}$ satisfies the open set
condition (OSC). This extends the well-known result for the self-similar case
that the OSC is equivalent to $K$ having positive Hausdorff measure
$\mathcal{H}^s$ for a suitable $s$. Furthermore, we relate the exact value of
pseudo Hausdorff measure $\mathcal{H}_w^s(K)$ to a notion of upper $s$-density
with respect to the pseudo norm $w(x)$ associated with $A$ for the measure
$\mu=\lim\limits_{M\to\infty}\sum\limits_{d_0,\dotsc,d_{M-1}\in\mathcal{D}}\delta_{d_0
+ Ad_1 + \dotsb + A^{M-1}d_{M-1}}$ in the case that $\#\mathcal{D}\le\lvert\det
A\rvert$.
