In a nutshell, no. Brita filters are intended to remove waterborne contaminants such as chemicals as well as sediment.
They are not designed to “purify” water or remove biological nasties like giardia.
So, get a filter that is intended for use in the backcountry or while camping. Consider the MSR Hyperflow ($100), which has an extremely high flow rate (three liters per minute) and easily attaches to a Nalgene-type bottle.
It eliminates protozoa, bacteria, and sediment. It won’t catch viruses; for that, you’ll need a post-filtering add-on like MSR’s SweetWater Purifier Solution ($10).
If you’re camping with a group, another MSR product worth considering is the Autoflow Microfilter ($100).
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It operates on a gravity-fed system. Simply fill a compatible water bag with water, attach the filter, and hang it up.
The flow rate is close to two liters per minute, and it performs all of the functions of the Hyperflow. Without using a single pump stroke, you can have water for four minutes.
The Katadyn Base Camp Water Filter ($70) operates in the same manner. Katadyn also makes hand-held filters, such as the toughly built Katadyn Pocket Water Filter ($290).
You can also use an individual bottle filter system like Sawyer’s Four-Way Filter ($55), which comes with a bottle that holds 32 fluid ounces, for filtered water on the go. Its filter can be used with the bottle, allowing you to sip whenever you want.
You can also use it to build a gravity filter system or attach it to a faucet. However, it is best suited for a single user.