There is a good summer fare available with American Airlines and Finnair to Stockholm Bromma BMA airport at $607 round trip in July 2017. Ticket prices are about $649 from San Francisco to Stockholm BMA or Gothenburg GOT in Sweden for July departures.
The routing is American Airlines to either Chicago or New York for Finnair to Helsinki and Stockholm Bromma BMA, a secondary airport for Stockholm. Arlanda ARN is the main international airport for most USA flights. These tickets book in economy O class. You might earn more redeemable miles booking codeshare BA or Finnair AY flights for 25% flight miles (25% is 3,423 miles for LAX-JFK-HEL-BMA round trip) compared to 5 miles/$1 as AA flight numbers.
American Airlines Los Angeles LAX – Chicago ORD – Helsinki, Finland HEL – Stockholm Bromma BMA – Helsinki – JFK – LAX
(note: Google Flights seems to have a link glitch with AA since AA changed their flight display format. On AA page change destination to BMA to see $607 fares).
American Airlines/Finnair Los Angeles LAX – New York JFK – Helsinki, Finland HEL – Stockholm Bromma BMA – Helsinki – JFK – LAX
There are routing options with potential for overnight in Helsinki if that is desirable.