The message in Approximations by Mona Simpson is hinted at in the title and is linked to Melinda's personal struggle to find answers to her questions about who is really important in her life and who can make a difference. An approximation is something that meets a broad description of a desired thing or state of being but it falls short of the real thing. The theme develops as the reader watches Melinda in her efforts to connect with her father. The reader can relate to her dilemma as she makes compromise after compromise in all her relationships. She thinks that a relationship with her father will resolve her issues because that makes the most sense, and as a child, she misinterprets her father's intentions, mistaking them for genuine interest and love. At the same time, she overlooks the potential for a relationship with Jerry because that would apparently be too much of a compromise.
When linking the message in this and any story to the theme, it is important to understand that a "message" is a defined example of a theme. The theme is the larger issue and the standard against which the message is revealed. In this story, the theme relates to compromise (which is the bigger issue) and the message in this case is that, for Melinda, appearances can be deceiving. Thoughts of her ideal father dominate Melinda's life but it is only after disappointment and her eventual acceptance that her relationship with her father does not meet her expectations and never will that she opens her heart, leaving enough room so that a meaningful relationship with Jerry can develop. Jerry, given the opportunity, can be the ideal father-figure and he has the potential to meet her expectations so that she no longer has to settle for a relationship which is only surface deep.