I have a question on the paper "A note on sufficient conditions for no arbitrage" by Carr / Madan:

They say \sum_{i=1}^\infty q_{i,j} = 1. To me it seems this sums equals Q_{1,j} = S_0 - C_{1,j} / K_1 != 1. Do we have add an additional strike, possibly K = K_0 = 0 and attach the probability 1 - Q_{1,j} to it to complete the probability distribution?

Less important, do we have to put an additional condition on the sequence of strikes to ensure (C_{i,j} - C_{i+1,j}) / (K_{i+1}-K_i) tends to zero when i goes to infinity, like "there is an epsilon > 0 s.t. K_{i+1}-K_i > epsilon for all i".

I agree with your sum and agree that an additional (unstated) condition seems to be needed. Since they are doing a lattice version of the continuum problem, let's first look at that. Fixing the maturity T, the call value is [$]C(K)[$]. Breeden and Litzenberger say that [$]q(K) \equiv C_{KK}(K)[$] can be interpreted, under some conditions, as the pdf of finding the stock price at strike K on maturity. What are the conditions? If the stock price cannot actually reach 0, then for mass preservation we need

[$] 1 = \int_{0^+}^{\infty} C_{KK} \, dK = C_K |^{K=\infty}_{K=0^+} [$]

The upper end is not problematic: since C(K) is decreasing to 0, [$]C_K[$] must be decreasing to 0 as well.

But, at the lower end, we need the condition that

(*) [$]C_K(0^+) = -1[$].

In the Carr-Madan lattice version this would be the condition that

[$] 1 = \frac{C_0 - C_1}{K_1 - K_0} = \frac{S_0 - C_1}{K_1}[$].

So you need to assume that [$]C_1 = S_0 - K_1[$] as the lattice version of (*).