Ole and Mari Finnestad's great granddaughter Elma Mathilda Johnson was born January 17, 1892 in Chicago, Illinois.
Cook County had both a birth certificate and a birth register. These are the only documents I've come across that list her name as Elma Mathilda. By the time the family had moved to Iowa, Elma was known as Lillie, which was probably a nickname for Mathilda.
Note the misspelling of Johnson as "Johanson" on the birth certificate.
Page 1 of the birth register lists the child's name, the sequence the child represents borne to the mother, address, nationality and age of the parents (though the mother's age of 18 was cut off). Elma is listed 12th from the bottom.
Page 2, which includes the mother's maiden name, Lizzie Knutson, as well as the father, Lambert Johnson (spelled incorrectly as Lamberth Johanson). The name of the doctor is listed in the column on the right.
Misspellings can throw you off, and considering I wasn't aware that Lillie was born with the name Elma Mathilda, there are often enough clues to ensure you have the right person.
Lambert's age in January 1892 was 21; he would turn 22 in April. Lizzie would turn 19 twelve days after the birth of her daughter, which matches the birth records.
I am not sure whether the Johnson family ever lived in Chicago, though Lambert's brother was living in Chicago by the time of the 1900 US census. It's possible the family was visiting Jacob when Lillie was born. Lillie's brother Truman was born in Rochelle, Illinois, so I would guess the family had lived in Rochelle prior to moving to Iowa.