app.rb

require 'sinatra/base'
require 'json'

class MyApp < Sinatra::Base
  before    { content_type "application/json;charset=utf-8" }
  not_found { JSON.dump("error" => "Not Found") }
end

Gemfile

source 'https://rubygems.org'

gem 'rack-test'
gem 'sinatra'

Gemfile.lock

GEM
  remote: https://rubygems.org/
  specs:
    rack (1.4.1)
    rack-protection (1.2.0)
      rack
    rack-test (0.6.1)
      rack (>= 1.0)
    sinatra (1.3.2)
      rack (~> 1.3, >= 1.3.6)
      rack-protection (~> 1.2)
      tilt (~> 1.3, >= 1.3.3)
    tilt (1.3.3)

PLATFORMS
  ruby

DEPENDENCIES
  rack-test
  sinatra

output

➜  sinatra_example  RACK_ENV=test ruby test.rb
Run options: 

# Running tests:

.

Finished tests in 0.019838s, 50.4083 tests/s, 100.8166 assertions/s.

1 tests, 2 assertions, 0 failures, 0 errors, 0 skips
➜  sinatra_example  ruby test.rb 
Run options: 

# Running tests:

F

Finished tests in 0.022823s, 43.8154 tests/s, 87.6309 assertions/s.

  1) Failure:
test_not_found(MyAppTest) [test.rb:15]:
<"application/json;charset=utf-8"> expected but was
<"text/html;charset=utf-8">.

1 tests, 2 assertions, 1 failures, 0 errors, 0 skips

test.rb

require 'test/unit'
require 'rack/test'
require_relative 'app'

class MyAppTest < Test::Unit::TestCase
  include Rack::Test::Methods

  def app
    MyApp.new
  end

  def test_not_found
    get '/not_found'
    assert_equal 404, last_response.status
    assert_equal 'application/json;charset=utf-8', last_response.headers['Content-Type']
  end
end