In the Lost series, when Kate's past life is shown where she marries a policeman, while being a fugitive. How is it possible that her husband had not come across Kate's profile as she was a highly dangerous criminal?
It's hard to answer this. She had a fake identity ("Monica"). She met him posing as this Monica and fell in love with him. He knew nothing about her former life, and like most people probably wasn't interested in doing thorough digging on her past life. When we meet people, we tend to believe what they tell us, rather than studiously examining their past life to see if what they say is true.
Of course, over time, most people's recounts of their past experiences are shown to be true, through photographs, or reunions with old friends, or knowledge of past experiences.
This is what caught up to Kate. As time passed, and she had to deal with a pregnancy scare and an ability to go on honeymoon as she didn't have a valid passport, she realised she couldn't continue with her husband. She confessed to him, having drugged him so he'd fall unconscious, then left him (the episode I Do)
So to specifically answer your question, it appears her husband was in love with her and simply took her at face value when she explained her fictional past. When he started to suspect things weren't completely right, she left him.