Conditioning on both sigma algebras and events

Question

I've recently come across the following notation: given independent discrete random variables $I_1,\ldots,I_k$ on a probability space and another random variable $W$ defined on it, such that $\{W=k\}\in\sigma(I_1,\ldots,I_k)$, the following conditional probability is well defined: $\mathbb{P}(A|I_1,I_2,\ldots,I_k,W=k)$ for some event $A$. Now, I am familiar with conditioning on the $\{I_j\}$ alone (that is on $\sigma(I_1,\ldots,I_k)$) and on events alone (that is on $(W=k)$ alone), but how does this mixed case arise? I simply wonder what its definition is. What role does the measurability of $(W=k)$ plays a in this definition? I know that since the random variables are discrete, we could just work with $\mathbb{P}(A|I_1=i_1,I_2=i_2,\ldots,I_k=i_k,W=k)$ and probably leave it at that. But then the question would arise anyway: if the $I_k$ were not discrete, would this mixed conditional probability still be well-defined? My attempt was to interpret the expression as $\mathbb{E}[\mathbb{P}(A|I_1,\ldots,I_k)|W=k]=\mathbb{E}[\mathbb{E}(1_A|I_1,\ldots,I_k)|W=k]=\mathbb{P}(A|W=k)$, by the definition of conditional expectation and measurability in the hypothesis, but I am not sure this is the correct interpretation of the expression. Thank you for any insight you might be able to share. Link to the paper: https://scholar.google.com/scholar_url?url=https://projecteuclid.org/journals/annals-of-probability/volume-28/issue-3/Sum-the-odds-to-one-and-stop/10.1214/aop/1019160340.pdf&hl=en&sa=T&oi=ucasa&ct=ufr&ei=N_BHZMbTMqiI6rQPlL-HGA&scisig=AJ9-iYv9tgIfEcKmKJcKwq3hHQ-Z. (second line of p.3).