pr.probability - Difference between Beta Process and Dirichlet process

I'm trying to understand the definition of a Beta process, as given in the paper:
www.ece.duke.edu/~lcarin/Paisley_BP-FA_ICML.pdf

The problem is that from the definition it follows that every Dirichlet process is also a beta process, which seems, ahm, wrong. Can you help me figure out what I don't understand?

This is the definition from the paper:
"Let $H_0$ be a continuous probability measure and α a positive scalar. Then for all disjoint, infinitesimal partitions, $B_1 ,ldots, B_K$, the beta
process is generated as follows,
$H(B_k) sim Beta(alpha H_0 (B_k ), alpha(1 − H_0(B_k )))$

with K → ∞ and $H_0(B_k)$ → 0 for $k = 1,ldots,K$. This process is denoted $H sim BP(alpha H_0 )$."

This is the definition for a Dirichlet Process (DP):

If $X sim DP(alpha H_0)$ where $alpha$ is a scalar and $H_0$ is a probability distribution, then for every finite partition $A_1,ldots,A_K$ it follows that
$(X(A_1),ldots,X(A_K)) sim DIR(alpha H_0(A_1), ldots,alpha H_0(A_K))$.

So let's assume that I have $Xsim DP(alpha H_0)$. Given any partition $B_1 ,ldots, B_K$, and any $k = 1 ldots K$, I can define a partition $A_1 = B_k, A_2 = Omega -B_k$ and from the DP definition it follows that

$(X(A_1),X(A_2)) sim DIR(alpha H_0(A_1), alpha H_0(A_2))$
which is equivalent to saying that
$X(B_k) sim Beta(alpha H_0(B_k), alpha(1-H_0(B_k)))$

hence $Xsim BP(alpha H_0)$. Where is my mistake?