You did not state how much research you had done on other family members, but here summarized is the 1910 census entry you found. Along with the 1900 census entry (listing her other children) and Martin's 1920 census entry (listing his wife and sons).
To keep these entries short, I will post several in answer to your questions, and also send long files off-list.
1910 census (3 people living in household):
BERG MARTIN T 29 M W NORW IL COOK 27-WD CHICAGOSeries: T624 Roll: 270 Page: 238 A
2250 Kimball ** NOTE: this is post 1909 address and street, so building may still exist and photo may be on Cook County Assessor's website!
Head: Martin T, 29, single, Norw, Electricianâ€“Installing telephones, emig 1888
Mother: Petra M, 69, widowed, Norw, emig 1888; mother of 14/4 living
Cousin: Einar G. Gronseth, 23, Norw, Clerk â€“ Post Office, emig 1892
1900 Census (don't know if you had this already)
BERG PETER M 58 M W NORW IL COOK 15-WD CHICAGO
(NOTE: transcription error - entry is Petra M Berg, Female)
Series: T623 Roll: 265 Page: 282 A
395 W Fullerton **NOTE: This is a pre-1909 street number so address number MUST be converted to post 1909 value (using website from Chicago Historical Society/Chicago History Museum); Fullerton is a major East-West street and retained its name!
Head: BERG Petra M, female, Widowed, Apr 1842, 58, Norway, emig 1890, mother 14/6 living
Son: Berg Peter M, male, Single, Nov 1868, 31, Norway, emig 1889, Na, Machinist
Son: Berg Einar, male, Single, Jan 1879, 21, Norway, emig 1889, Al, Inspector â€“ Wood
Son: Berg Martin, male, Single, Jan 1881, 19, Norway, emig 1890, Al, Machinist (Harvestor Co)
*** Note: Harvestor Co was probably International Harvestor (successor co: Navistar), a major employer in Chicago in the 1900s - my grandfather worked there 49 years!
Dau: Berg Marie, female, Single, June 1883, 16, Norway, emig 1890, Al, --
Nephew: Gronseth Einar, male, Single, March 1887, 13, Norway, emig 1892, Al, Student
1920 Census - only found son Martin (so far):
BERG MARTEN 39 M W NORW IL COOK 33-WD CHICAGO
Series: T625 Roll: 352 Page: 179 B
2208 Keystone ** This street and number should also exist on modern maps
Martin â€“ a telephone tester
We can assume that Petra was either a widow when she emigrated with children (or joined children already in Chicago). Or her husband died between 1888-1900 (do you have a name?)!
Also have you located the Digitalarkivet records for emigration or 1865 census for the family, or Petra premarried?