In this question, I assume you refer to Joyce Carol Oates' version of "The Lady With the Pet Dog," rather an Chekov's version. While Oates' version is based upon Chekov's story, it is very different in several ways. One of the aspects that is changed in Oates' version is the importance of the woman in the story, rather than the development of a male protagonist; and the setting, which influences the plot development, is the 1970s, when women's roles within society were changing, as opposed to Chekov's late nineteenth century Russia where a woman had little or no rights, and rarely a unique identity other than the one cast on her by society.
Besides the setting in time and place, a major change in this version of the story is the reader's ability to follow the life and emotions of Anna. She is married and ends up having an affair with an unnamed man she meets while vacationing without her family in New England. As the story goes on, Anna knows she has no future with her lover, but is not in love with her husband, and is torn in two directions—neither of which are satisfactory. Anna even tries to kill herself. She has poor self-esteem, judging herself harshly for her adulterous affair.
Even though Anna ends the relationship, by chance she later meets her lover at a concert and they resume their affair. By the end of the story, Anna seems to have found herself.
The [third] section [of the story] ends, however, with Anna's joyous vision of her affair as a true marriage; her resulting happiness surprises her lover.
This simply summarizes the story. The themes Oates' investigates were relevant in the 1970s, and are still relevant for some today.
Anna's struggle in the male-dominated society of the United States is shared by other female protagonists in Oates's early work: their self-images depend on men, not on themselves or other women. Although she also considers it absurd, Anna feels she needs men to save her, and she behaves accordingly. Like an addict, she looks for salvation in the very things that hurt her.
The eNotes.com study of the themes within this story draw the reader's attention to the "male-dominated" society. This, then, becomes a story that also has deep threads of feminism woven within as Anna's character represents the inner-struggle of women to find out who they are (with or without a man). Anna battles with the sense that she needs a man to be whole or fulfilled: she considers the need "absurd." At the end it would seem that Anna has finally found joy in the relationship she has accepted with her lover...but still she is depending upon a man for happiness. However, if her decision is based upon an independent choice rather than giving in to societal expectations that she be with a man, or her own need, it would seem not to be a mistake.
Another element that makes this a feminist piece are the choices that Oates' female character are generally given in her stories: to go mad or to find a path to "self-empowerment." Anna travels both paths. In terms of madness, she tries to kill herself. However, her willingness to concentrate more on being happy with her lover rather than guilty with him, or miserable with her husband, shows that Anna is starting to redefine the boundaries of her world, seeming to make a choice she is happy with.
There is no guarantee that Anna will be happy with her lover indefinitely, but this is a realistic aspect of any relationship between two people, married or lovers.