This question pertaining to Measure for Measure may depend on precisely what you mean by "upper world" and "lower world." In Greek and Roman mythology, the upper world is Earth and the abode of humankind, while the lower world is the habitation of the dead and is ruled by Hades and Persephone. On the other hand, in Christian terminology, the upper world is the spiritual world and the realm of angels and God, while the lower world is Earth and the abode of humankind.

If you are expressing your question in terms of mythology, then the lower world of death might be seen as illuminating the upper world of humankind by establishing a measuring stick for the valuation of life: escape from death is worth what cost to life? This is the question that the Duke's test and Angelo's villany forces Isabella to decide on. Her choice establishes her answer as stating that spiritual values exceed physical values (as a novice of the Church, chasity is a spiritual promise not a physical condition).

If you are speaking in terms of Christian theology, then the lower world of humankind might be seen as illuminating the upper world of spirituality and the abode of God by--well, actually, the same scenario with more applications: the physical world is secondary to spiritual truths. Again, Claudio's impending human death is less important than a spiritual truth of unity with God, or rather the promise of future unity with God. Another application is that selfish and carnal behavior, such as Angelo's (and Lucio's) crumbles when measured in the face of truth, true justice, and righteousness.

In Measure for Measure, how is the story of the upper world illuminated by the lower world?

I'm not quite sure what you're asking, but I have an inkling, so here goes:  The "upper world" and "lower world," in the world of the play could be better expressed as plot and subplot.

There are often, in Shakespeare's comedies, high born characters that populate the central plot of the play -- in this case, Isabella, Angelo, and to some degree the Duke -- and lower born characters that populate the subplot of the play.  The subplot often mirrors the main plot, giving a sort of commentary on the plot.

The main plot of this play revolves around a woman, Isabella, who has pledged to become a nun pleading for her brother's life (mercy) to the Deputy that is in charge in the Duke's absence, Angelo.  Angelo agrees to release her brother in exchange for a sexual tryst with Isabella.

The simplest illumination provided of this plot by the subplot characters, is that the subplot characters are corrupt, licentious people living outside the laws of "decent" society.  These characters could be said to demonstrate Angelo's real demeanor, and the sort of life he would act out behind closed doors.  The subplot characters, however, do so out in the open.