You can, but most players don't. If the helmet is too big, wearing your cap under it is a good way to make it fit better though.
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I have never heard baseball caps called that but there is a line of baseball caps marketed by New Era that are called '59FIFTY'.
some where "skull caps" which is like a compression type of material but in high school its a rule that you can't wear anything under your helmet
The baseball cap was originally invented in Europe somewhere in the west. The first baseball caps had a strap of cow hide to keep them on. It is thought that they were first made in the 1890's.
For games on July 4 weekend, Major League Baseball Clubs will wear special "Stars & Stripes" caps, which have the American flag etched into the team's logo and are off-white in color with red or blue brims (the Toronto Blue Jays hat will incorporate a Maple Leaf design instead of the "Stars & Stripes"). Major League Baseball Properties will donate 100% of the proceeds it receives from the sale of the caps to Welcome Back Veterans, a program which addresses the needs of returning American Veterans and their families.
In 1920, Mule Watson played in 1 game for the Boston Red Caps. He had 1 at bats, getting 0 hits, for a .000 batting average, with 0 runs batted in. He was walked 0 times. He struck out 0 times.