The Besicovitch-Federer structure theorem enables us to make the following definition:



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> Suppose $k < n$, $E \subset \mathbf{R}^n$, and $0 < \mathcal{H}^k(E) < \infty$. We say $E$ is **purely unrectifiable**, provided that $\mathcal{H}^k(\pi\_K(E))= 0$ for almost every $k$-plane $K$.
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Many examples of such purely unrectifiable sets are known (see for example, Frank Morgan's book on geometric measure theory). 


We can now try to make the following definition:



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> Suppose $d \leq k < n$, $F \subset \mathbf{R}^n$, and $0 < \mathcal{H}^d(F) < \infty$. We say $F$ is **$(d, k)$-unrectifiable**, provided that $\mathcal{H}^d(\pi\_K(F))= 0$ for almost every $k$-plane $K$.
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Thus, in my terminology $(k,k)$-unrectifiability is what is also called pure unrectifiability. 


Is anything known about the case $d < k$? It doesn't seem obvious that such sets $F$ can't exist. 



Generalizing a notion of purely unrectifiable sets
Actually quite a lot is known about such sets. You can find many results in Chapter 9 of Mattila's *Geometry of Sets and Measures in Euclidean Spaces*. Here are some examples:



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> **Theorem 1.** If $0<d\leq 1$, then there is a set $F\subset\mathbb{R}^2$ such that $0<H^d(F)<\infty$, but projections on
>  every line have $H^d$ measure zero.
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That implies that a set can be $(d,k)$-unrectifiable and still have positive $H^d$-measure. However, we also have



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> **Theorem 2.** If $F\subset\mathbb{R}^n$ is Borel and $d\leq k<n$ and $\dim F\leq k$, then $\dim \pi\_K(F)=\dim F$ for allmost all $k$-planes
>  $K$.
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Here $\dim$ stands for the Hausdorff dimension.


In particular, if in addition $0<H^d(F)<\infty$ (as in your assumption), then $\dim\pi\_K(F)=d$ for alsmost all $k$-planes $K$. However, Theorem 1 shows in general you cannot expect $H^d(\pi\_K(F))$ to be positive.


