Kappa-filtration ($(A_{\alpha }{)}_{\alpha <\kappa }$)

For a set $A$ of size an uncountable regular cardinal $\kappa$, an increasing sequence $(A_{\alpha }{)}_{\alpha <\kappa }$ with union $A$, each $|A_{\alpha }|<\kappa$, and $A_{\delta }={\bigcup }_{\alpha <\delta }{A}_{\alpha }$ at every nonzero limit stage.